Maximum Degree

Multicast communication in a wireless ad-hoc network can be established using a tree that spans the multicast sender and receivers as well as other intermediate nodes. If the network is modelled as a graph, the multicast tree is a Steiner tree, the multicast sender and receivers correspond to terminals, and other nodes participating in the tree are Steiner nodes. As Steiner nodes are nodes that participate in the multicast tree by forwarding packets but do not benefit from the multicast, it is a natural objective to compute a tree that minimizes the total cost of the Steiner nodes. We therefore consider the problem of computing, for a given nodeweighted graph and a set of terminals, a Steiner tree with Steiner nodes of minimum total weight. For graph classes that admit spanning trees of maximum degree at most d, we obtain a 0.775d-approximation algorithm. We show that this result implies a 3.875-approximation algorithm for unit disk graphs, an O(1/α 2 )-approximation algorithm for α-unit disk graphs, and an O(λ)-approximation algorithm for (λ + 1)claw-free graphs.

Multicast communication in a wireless ad-hoc network can be established using a tree that spans the multicast sender and receivers as well as other intermediate nodes. If the network is modelled as a graph, the multicast tree is a Steiner tree, the multicast sender and receivers correspond to terminals, and other nodes participating in the tree are Steiner nodes. As Steiner nodes are nodes that participate in the multicast tree by forwarding packets but do not benefit from the multicast, it is a natural objective to compute a tree that minimizes the total cost of the Steiner nodes. We therefore consider the problem of computing, for a given nodeweighted graph and a set of terminals, a Steiner tree with Steiner nodes of minimum total weight. For graph classes that admit spanning trees of maximum degree at most d, we obtain a 0.775d-approximation algorithm. We show that this result implies a 3.875-approximation algorithm for unit disk graphs, an O(1/α 2 )-approximation algorithm for α-unit disk graphs, and an O(λ)-approximation algorithm for (λ + 1)claw-free graphs.

A subset S of the vertex set of a graph G is a dominating set if every vertex in the complement of S is adjacent to some vertex of S. The minimum cardinality among all minimal dominating sets is the domination number of the graph G. For a minimal dominating set D of G if the graph does not have any isolated vertices, then V -D contains a dominating set. Such a dominating set in V -D is called an inverse dominating set of G and the minimum cardinality among all inverse dominating sets is called the inverse domination number. This paper surveys existing results on inverse dominating sets, inverse domination number and lists some open problems on these concepts. The Kulli-Sigarkanti conjecture on inverse domination number is settled.

Efficient algorithms for computing routing tables should take advantage of the particular properties arising in large scale networks. Two of them are of particular interest: low (logarithmic) diameter and high clustering coefficient. High clustering coefficient implies the existence of few large induced cycles. Considering this fact, we propose here a routing scheme that computes short routes in the class of kchordal graphs, i.e., graphs with no induced cycles of length more than k. In the class of k-chordal graphs, our routing scheme achieves an additive stretch of at most k -1, i.e., for all pairs of nodes, the length of the route never exceeds their distance plus k -1. In order to compute the routing tables of any n-node graph with diameter D we propose a distributed algorithm which uses messages of size O(log n) and takes O(D) time. The corresponding routing scheme achieves the stretch of k -1 on k-chordal graphs. We then propose a routing scheme that achieves a better additive stretch of 1 in chordal graphs (notice that chordal graphs are 3-chordal graphs). In this case, the distributed computation of the routing tables takes O(min{∆D, n}) time, where ∆ is the maximum degree of the graph. Our routing schemes use addresses of size log n bits and local memory of size 2(d -1) log n bits per node of degree d.

A subset S of the vertex set of a graph G is a dominating set if every vertex in the complement of S is adjacent to some vertex of S. The minimum cardinality among all minimal dominating sets is the domination number of the graph G. For a minimal dominating set D of G if the graph does not have any isolated vertices, then V -D contains a dominating set. Such a dominating set in V -D is called an inverse dominating set of G and the minimum cardinality among all inverse dominating sets is called the inverse domination number. This paper surveys existing results on inverse dominating sets, inverse domination number and lists some open problems on these concepts. The Kulli-Sigarkanti conjecture on inverse domination number is settled.

The edge-reconstruction number ern(G) of a graph G is equal to the minimum number of edge-deleted subgraphs G-e of G which are sufficient to determine G up to isomorphism. Building upon the work of Molina and using results from computer searches by Rivshin and more recent ones which we carried out, we show that, apart from three known exceptions, all bicentroidal trees have edge-reconstruction number equal to 2. We also exhibit the known trees having edge-reconstruction number equal to 3 and we conjecture that the three infinite families of unicentroidal trees which we have found to have edge-reconstruction number equal to 3 are the only ones.

Berman and Schnitger (10) gave a randomized reduction from approximating MAX- SNP problems (24) within constant factors arbitrarily close to 1 to approximating clique within a factor of n (for some ). This reduction was further studied by Blum (11), who gave it the name randomized graph products. We show that this reduction can be made deterministic (derandomized), using random

-representation of a graph G consists of a collection of subtrees {Sv|v ∈ V (G)} of a tree T , such that (i) the maximum degree of T is at most h, (ii) every subtree has maximum degree at most s, and (iii) there is an edge between two vertices in the graph if and only if the corresponding subtrees in T have at least t vertices in common. For example, chordal graphs correspond to [∞, ∞, 1] = [3, 3, 1] = [3, 3, 2] graphs (notation of ∞ here means that no restriction is imposed). We investigate the complete bipartite graph K2,n and prove new theorems characterizing those K2,n graphs that have an (h, s, 2)-representation and those that have an (h, s, 3)-representation. We characterize [3, 2, 4] graphs as equivalent to the 4-flower-free [2, 4, 4] graphs and give a recognition algorithm for [2, 3, 4] graphs. Based on these characterizations, we present new results that confirm that weakly chordal graphs, as opposed to chordal graphs, can not be characterized within the [h, s, t] framework. Furthermore, we show a hierarchy of families of graphs between chordal and weakly chordal within the [h, s, t] framework.

We first present new structural properties of a two-pair in various graphs. A twopair is used in a well-known characterization of weakly chordal graphs. Based on these properties, we prove the main theorem: a graph G is a weakly chordal (K 2,3 , 4P 2 , P 2 ∪ P 4 , P 6 , H 1 , H 2 , H 3 )-free graph if and only if G is an edge intersection graph of subtrees on a tree with maximum degree 4. This characterizes the so called graphs. The proof of the theorem constructively finds the representation. Thus, we obtain an algorithm to construct an edge intersection model of subtrees on a tree with maximum degree 4 for such a given graph. This is a recognition algorithm for graphs.

An (h, s, t)-representation of a graph G consists of a collection of subtrees of a tree T , where each subtree corresponds to a vertex in G, such that (i) the maximum degree of T is at most h, (ii) every subtree has maximum degree at most s, (iii) there is an edge between two vertices in the graph G if and only if the corresponding subtrees have at least t vertices in common in T . The class of graphs that have an (h, s, t)-representation is denoted by [h, s, t]. It is well known that the class of chordal graphs corresponds to the class [3, 3, 1]. Moreover, it was proved by Jamison and Mulder that chordal graphs correspond to orthodox-[3, 3, 1] graphs defined below. In this paper, we investigate the class of [h, 2, t] graphs, i.e., the intersection graphs of paths in a tree. The [h, 2, 1] graphs are also known as path graphs [F. Gavril, A recognition algorithm for the intersection graphs of paths in trees, Discrete Math. 23 (1978) 211-227] or VPT graphs [M.C. Golumbic, R.E. Jamison, Edge and vertex intersection of paths in a tree, Discrete Math. 55 (1985) 151-159], and [h, 2, 2] graphs are known as the EPT graphs. We consider variations of [h, 2, t] by three main parameters: h, t and whether the graph has an orthodox representation. We give the complete hierarchy of relationships between the classes of weakly chordal, chordal, [h, 2, t] and orthodox-[h, 2, t] graphs for varied values of h and t.

In this paper,  we investigate the structural properties of trees and bipartite graphs through the lens of topological indices and combinatorial graph theory. We focus on the First and Second Hyper-Zagreb indices, $HM_1(G)$ and $HM_2(G)$, for trees $T \in T(n, \Delta)$ with $n$ vertices and maximum degree $\Delta$. Key propositions demonstrate that the presence of end-support or support vertices of degree at least three, distinct from a vertex of maximum degree, implies the existence of another tree $T' \in T(n, \Delta)$ with strictly smaller Hyper-Zagreb indices. These results are extended to exponential forms, highlighting the influence of high-degree vertices. Additionally, we explore structural characterizations of trees via degree sequence majorization and $S$-order, establishing conditions for the first and last trees in specific classes. For bipartite graphs, we examine equitable coloring, cycle lengths, and $k$-redundant tree embeddings, supported by theorems on connectivity and minimum degree constraints. The paper also addresses the independence number of bipartite graphs, exterior covers, and competition numbers of complete $r$-partite graphs, providing bounds and structural insights. Finally, we discuss Markov-chain algorithms for generating bipartite graphs and tournaments with prescribed degree sequences, analyzing their mixing times and convergence properties. These results contribute to the understanding of extremal properties and combinatorial structures in graph theory, with applications in chemical graph theory and network analysis.

We study the problem of packing element-disjoint Steiner trees in graphs. We are given a graph and a designated subset of terminal nodes, and the goal is to find a maximum cardinality set of elementdisjoint trees such that each tree contains every terminal node. An element means a non-terminal node or an edge. (Thus, each non-terminal node and each edge must be in at most one of the trees.) We show that the problem is APX-hard when there are only three terminal nodes, thus answering an open question. Our main focus is on the special case when the graph is planar. We show that the problem of finding two element-disjoint Steiner trees in a planar graph is NP-hard. Similarly, the problem of finding two edge-disjoint Steiner trees in a planar graph is NP-hard. We design an algorithm for planar graphs that achieves an approximation guarantee close to 2. In fact, given a planar graph that is k elementconnected on the terminals (k is an upper bound on the number of element-disjoint Steiner trees), the algorithm returns k 2 -1 element-disjoint Steiner trees. Using this algorithm, we get an approximation algorithm for the edge-disjoint version of the problem on planar graphs that improves on the previous approximation guarantees. We also show that the natural LP relaxation of the planar problem has an integrality ratio approaching 2.

An L(2; 1)-coloring of a graph G is a coloring of G's vertices with integers in {0; 1; : : : ; k} so that adjacent vertices' colors di er by at least two and colors of distance-two vertices di er. We refer to an L(2; 1)-coloring as a coloring. The span (G) of G is the smallest k for which G has a coloring, a span coloring is a coloring whose greatest color is (G), and the hole index (G) of G is the minimum number of colors in {0; 1; : : : ; (G)} not used in a span coloring. We say that G is full-colorable if (G) = 0. More generally, a coloring of G is a no-hole coloring if it uses all colors between 0 and its maximum color. Both colorings and no-hole colorings were motivated by channel assignment problems. We deÿne the no-hole span (G) of G as ∞ if G has no no-hole coloring; otherwise (G) is the minimum k for which G has a no-hole coloring using colors in {0; 1; : : : ; k}. Let n denote the number of vertices of G, and let be the maximum degree of vertices of G. Prior work shows that all non-star trees with ¿ 3 are full-colorable, all graphs G with n = (G) + 1 are full-colorable, (G) 6 (G) + (G) if G is not full-colorable and n ¿ (G) + 2, and G has a no-hole coloring if and only if n ¿ (G) + 1. We prove two extremal results for colorings. First, for every m ¿ 1 there is a G with (G) = m and (G) = (G) + m. Second, for every m ¿ 2 there is a connected G with (G) = 2m, n = (G) + 2 and (G) = m.

Let G be an n-vertex graph that contains linearly many cherries (i.e., paths on 3 vertices), and let c be a coloring of the edges of the complete graph Kn such that at each vertex every color appears only constantly many times. In 1979, Shearer conjectured that such a coloring c must contain a properly colored copy of G. We establish this conjecture in a strong form, showing that it holds even for graphs G with O(n 4/3 ) cherries and moreover this bound on the number of cherries is best possible up to a constant factor. We also prove that one can find a rainbow copy of such G in every edge-coloring of Kn in which all colors appear bounded number of times. Our proofs combine a framework of Lu and Székely for using the lopsided Lovász local lemma in the space of random bijections together with some additional ideas.

We prove that the acyclic chromatic number of a graph with maximum degree ∆ is less than 2.835∆ 4/3 + ∆. This improves the previous upper bound, which was 50∆ 4/3 . To do so, we draw inspiration from works by Alon, McDiarmid and Reed and by Esperet and Parreau.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

We are interested here in homomorphisms of undirected graphs and relate them to graph degeneracy and bounded degree property. Our main result reads that both in the abundance of counterexamples and from the complexity point of view the Brooks's theorem and ÿrst ÿt algorithm are the 'only' easy cases.

The minimum cost homomorphism problem is a natural optimization problem for homomorphisms to a fixed graph H. Given an input graph G, with a cost associated with mapping any vertex of G to any vertex of H, one seeks to minimize the sum of costs of the assignments over all homomorphisms of G to H. The complexity of this problem is well understood, as a function of the target graph H. For bipartite graphs H, the problem is polynomial time solvable if H is a proper interval bigraph, and is NP-complete otherwise. In many applications, the costs may be assumed to be the same for all vertices of the input graph. We study the complexity of this restricted version of the minimum cost homomorphism problem. Of course, the polynomial cases are still polynomial under this restriction. We expect the same will be true for the NP-complete cases, i.e., that the complexity classification will remain the same under the restriction. We verify this for the class of trees. For general graphs H, we prove a partial result: the problem is polynomial if H is a proper interval bigraph and is NP-complete when H is not chordal bipartite.

The minimum value of a monotonic increasing function de-"rined on G partially ordered set S is assumed on the set of rninimai points of S. This observation is used to devise an efficient method for finding the m smallest functional values of monotonic functions defined on ordered pairs of positive integers. The method is easily extended to include monotonic functions defined on ordered n-tuples. Included is a FORTRAN program which was written to implement the procedure for a cert'oin important case. t ~, I n t roduei;iOil :\ general procedure is discussed for finding, in correct order, the m smallest values of a furietion defined on c,c&,rod pairs of positive integers when the given function .> ];i<:r('ash~g in each variable. Such functions al:id the need ~) ,:,rdor lheir vahies arise, Rir example, when differential :,igv svahie problems are solved by separation of variables ] 9 "'

We consider the problem of generating all maximal cliques in an unit disk graph. General algorithms to find all maximal cliques are exponential, so we rely on a polynomial approximation. Our algorithm makes use of certain key geographic structures of these graphs. For each edge, we limit the set of vertices that may form cliques with this as the longest edge. We then consider several characteristic shapes determined by that edge, and prove that all cliques having this as the longest edge, are included in one of the sets of vertices contained in these shapes. Our algorithm works in O(m∆ 2 ) time and generates O(m∆) cliques, where m is the number of edges in the graph and ∆ is its maximum degree. We also provide a modified version of the algorithm which improves the performance in many cases, albeit without affecting the worst case running time.

A restricted version of interactive proofs, denoted as Arthur-Merlin games, was defined by Babai in [B]. The difference is in that the verifier is restricted to" send messages Xj such that the concatenation x lX 2 ••• x, is the "secret" random string r. The verifier and prover in such protocols were denoted as Arthur (A) and Merlin (M) respectively, and the protocol is denoted as an AM protocol. AM [g (n)] and AM are dermed similarly to IP [g (n)] andIP.

The Frequency Assignment Problem (FAP) in radio networks is the problem of assigning frequencies to transmitters exploiting frequency reuse while keeping signal interference to acceptable levels. The FAP is usually modelled by variations of the graph coloring problem. A Radiocoloring (RC) of a graph G(V, E) is an assignment function Λ : V → IN such that |Λ(u) -Λ(v)| ≥ 2, when u, v are neighbors in G, and |Λ(u) -Λ(v)| ≥ 1 when the distance of u, v in G is two. The discrete number of frequencies used is called order and the range of frequencies used, span. The optimization versions of the Radiocoloring Problem (RCP) are to minimize the span (min span RCP) or the order (min order RCP). In this paper, we deal with an interesting, yet not examined until now, variation of the radiocoloring problem: that of satisfying frequency assignment requests which exhibit some periodic behavior. In this case, the interference graph (modelling interference between transmitters) is some (infinite) periodic graph. Infinite periodic graphs usually model finite networks that accept periodic (in time, e.g. daily) requests for frequency assignment. Alternatively, they can model very large networks produced by the repetition of a small graph. A periodic graph G is defined by an infinite two-way sequence of repetitions of the same finite graph G i (V i , E i ). The edge set of G is derived by connecting the vertices of each iteration G i to some of the vertices of the next iteration G i+1 , the same for all G i . We focus on planar periodic graphs, because in many cases real networks are planar and also because of their independent mathematical interest. We give two basic results: • We prove that the min span RCP is P SP ACE-complete for periodic planar graphs.

We study the time needed for deterministic leader election in the LOCAL model, where in every round a node can exchange any messages with its neighbors and perform any local computations. The topology of the network is unknown and nodes are unlabeled, but ports at each node have arbitrary fixed labelings which, together with the topology of the network, can create asymmetries to be exploited in leader election. We consider two versions of the leader election problem: strong LE in which exactly one leader has to be elected, if this is possible, while all nodes must terminate declaring that leader election is impossible otherwise, and weak LE, which differs from strong LE in that no requirement on the behavior of nodes is imposed, if leader election is impossible. We show that the time of leader election depends on three parameters of the network: its diameter D, its size n, and its level of symmetry λ, which, when leader election is feasible, is the smallest depth at which some node has a unique view of the network. It also depends on the knowledge by the nodes, or lack of it, of parameters D and n.

We consider the fundamental problems of size discovery and topology recognition in radio networks modeled by simple undirected connected graphs. Size discovery calls for all nodes to output the number of nodes in the graph, called its size, and in the task of topology recognition each node has to learn the topology of the graph and its position in it. In radio networks, nodes communicate in synchronous rounds and start in the same round. In each round a node can either transmit the same message to all its neighbors, or stay silent and listen. At the receiving end, a node v hears a message from a neighbor w in a given round, if v listens in this round, and if w is its only neighbor that transmits in this round. If more than one neighbor of a node v transmits in a given round, there is a collision at v. We do not assume collision detection: in case of a collision, node v does not hear anything (except the background noise that it also hears when no neighbor transmits). The time of a deterministic algorithm for each of the above problems is the worst-case number of rounds it takes to solve it. Nodes have labels which are (not necessarily different) binary strings. Each node knows its own label and can use it when executing the algorithm. The length of a labeling scheme is the largest length of a label. Our goal is to construct short labeling schemes for size discovery and topology recognition in arbitrary radio networks, and to design efficient deterministic algorithms for each of these tasks, using these short schemes. It turns out that the optimal length of labeling schemes for both problems depends on the maximum degree ∆ of the graph. For size discovery, we construct a labeling scheme of length O(log log ∆) (which is known to be optimal, even if collision detection is available) and we design an algorithm for this problem using this scheme and working in time O(log 2 n), where n is the size of the graph. We also show that time complexity O(log 2 n) is optimal for the problem of size discovery, whenever the labeling scheme is of optimal length O(log log ∆). For topology recognition, we construct a labeling scheme of length O(log ∆), and we design an algorithm for this problem using this scheme and working in time O D∆ + min(∆ 2 , n) , where D is the diameter of the graph. We also show that the length of our labeling scheme is asymptotically optimal, by proving that topology recognition in the class of arbitrary radio networks requires labeling schemes of length Ω(log ∆).

In the effort to understand the algorithmic limitations of computing by a swarm of robots, the research has focused on the minimal capabilities that allow a problem to be solved. The weakest of the commonly used models is Asynch where the autonomous mobile robots, endowed with visibility sensors (but otherwise unable to communicate), operate in Look-Compute-Move cycles performed asynchronously for each robot. The robots are often assumed (or required to be) oblivious: they keep no memory of observations and computations made in previous cycles. We consider the setting when the robots are dispersed in an anonymous and unlabeled graph, and they must perform the very basic task of exploration: within finite time every node must be visited by at least one robot and the robots must enter a quiescent state. The complexity measure of a solution is the number of robots used to perform the task. We study the case when the graph is an arbitrary tree and establish some unexpected results. We first prove that there are n-node trees where Ω(n) robots are necessary; this holds even if the maximum degree is 4. On the other hand, we show that if the maximum degree is 3, it is possible to explore with only O( log n log log n ) robots. The proof of the result is constructive. Finally, we prove that the size of the team is asymptotically optimal: we show that there are trees of degree 3 whose exploration requires Ω( log n log log n ) robots.

A subset of vertices in a graph is called a total dominating set if every vertex of the graph is adjacent to at least one vertex of this set. A total dominating set is called minimal if it does not properly contain another total dominating set. In this paper, we study graphs whose all minimal total dominating sets have the same size, referred to as well-totally-dominated (WTD) graphs. We first show that WTD graphs with bounded total domination number can be recognized in polynomial time. Then we focus on WTD graphs with total domination number two. In this case, we characterize triangle-free WTD graphs and WTD graphs with packing number two, and we show that there are only finitely many planar WTD graphs with minimum degree at least three. Lastly, we show that if the minimum degree is at least three then the girth of a WTD graph is at most 12. We conclude with several open questions.

We consider a relaxation of the concept of well-covered graphs, which are graphs with all maximal independent sets of the same size. The extent to which a graph fails to be well-covered can be measured by its independence gap, defined as the difference between the maximum and minimum sizes of a maximal independent set in G. While the wellcovered graphs are exactly the graphs of independence gap zero, we investigate in this paper graphs of independence gap one, which we also call almost well-covered graphs. Previous works due to Finbow et al. and Barbosa et al. have implications for the structure of almost well-covered graphs of girth at least k for k ∈ {7, 8}. We focus on almost well-covered graphs of girth at least 6. We show that every graph in this class has at most two vertices each of which is adjacent to exactly 2 leaves. We give efficiently testable characterizations of almost well-covered graphs of girth at least 6 having exactly one or exactly two such vertices. Building on these results, we develop a polynomial-time recognition algorithm of almost well-covered {C3, C4, C5, C7}-free graphs.

The reload cost refers to the cost that occurs along a path on an edge-colored graph when it traverses an internal vertex between two edges of different colors. Galbiati et al. [1] introduced the Minimum Reload Cost Cycle Cover problem, which is to find a set of vertex-disjoint cycles spanning all vertices with minimum reload cost. They proved that this problem is strongly NP-hard and not approximable within 1/ǫ for any ǫ > 0 even when the number of colors is 2, the reload costs are symmetric and satisfy the triangle inequality. In this paper, we study this problem in complete graphs having equitable or nearly equitable 2-edge-colorings, which are edge-colorings with two colors such that where c i (v) is the set of edges with color i that is incident to v. We prove that except possibly on complete graphs with fewer than 13 vertices, the minimum reload cost is zero on complete graphs with nearly equitable 2-edge-colorings by proving the existence of a monochromatic cycle cover. Furthermore, we provide a polynomial-time algorithm that constructs a monochromatic cycle cover in complete graphs with an equitable 2-edgecoloring except possibly in a complete graph with four vertices. Our algorithm also finds a monochromatic cycle cover in complete graphs with a nearly equitable 2-edge-coloring except some special cases.

In this paper we consider list colouring of a graph G in which the sizes of lists assigned to different vertices can be different. We colour G from the lists in such a way that each colour class induces an acyclic graph. The aim is to find the smallest possible sum of all the list sizes, such that, according to the rules, G is colourable for any particular assignment of the lists of these sizes. This invariant is called the D1-sum-choice-number of G. In the paper we investigate the D1-sum-choice-number of graphs with small degrees. Especially, we give the exact value of the D1-sum-choice-number for each grid Pn Pm, when at least one of the numbers n, m is less than five, and for each generalized Petersen graph. Moreover, we present some results that estimate the D1-sum-choice-number of an arbitrary graph in terms of the decycling number, other graph invariants and special subgraphs.

A proper edge colouring of a graph with natural numbers is consecutive if colours of edges incident with each vertex form an interval of integers. The deficiency def (G) of a graph G is the minimum number of pendant edges whose attachment to G makes it consecutively colourable. In 1999 Giaro, Kubale and Małafiejski considered the deficiency of the Hertz graphs. In this paper we study the deficiency of graphs from much wider class, which we call the generalized Hertz graphs. We find the exact values of the deficiency of all graphs from this class. Our investigation confirms, in this class, the conjecture that the deficiency of an arbitrary graph is not greater than its order. Moreover, we describe necessary and sufficient conditions which guarantee that the generalized Hertz graph is consecutively colourable, and necessary and sufficient conditions which guarantee that such a graph is minimal consecutively non-colourable. Applying the last mentioned result, we give the generating function for the sequence whose specified elements represent numbers of the minimal generalized Hertz graphs that are not consecutively colourable. One can find (Petrosyan and Khachatrian, 2014) the sufficient condition for consecutive non-colourability of bipartite graphs constructed based on trees in which any two leaves are in an even distance. In the paper we show that the same condition is also necessary for generalized Hertz graphs.

A graph property is any class of simple graphs, which is closed under isomorphisms. Let H be a given graph on vertices v1, …, vn. For graph properties 𝒫1, …, 𝒫n, we denote by H[𝒫1, …, 𝒫n] the class of those (𝒫1, …, 𝒫n) ‐partitionable graphs G, with a corresponding vertex partition (V1, …, Vn), for which an edge {xi, xj} with xi∈Vi and xj∈Vj implies the existence of the edge {vi, vj} in the graph H. The problem of the unique description of a graph property 𝒫 in the form H[𝒫1, …, 𝒫n] is investigated for 𝒫, 𝒫1, …, 𝒫n being from the class La of all graph properties closed under taking disjoint unions and subgraphs. The unique factorization theorems obtained in the paper generalize known results of this type bringing together ∘ ‐reducibility over La and ∨ ‐reducibility in the lattice (La, ⊆). There is also offered a new insight into the modular decomposition tree for a graph. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 48–64, 2009

Connected Dominating Sets (CDSs) can serve as virtual backbones for wireless networks. A smaller virtual backbone (minimum size CDS) incurs less communication overhead. Unfortunately, computing a minimum size CDS is NP-hard and thus many algorithms were developed which concentrates on how to construct smaller CDSs. Aiming for minimum number of nodes in CDS, results in CDSs that are highly unstable. Here we present four CDS algorithms which are constructed based on strategy, density, Threshold Neighborhood Distance Ratio (TNDR) and velocity. Any node in the network can be a strategic node depending upon the application and need and will be selected as the starting node instead of the node with highest density which concentrates on minimum size CDS. Simulation methods are used to establish the efficiency of the proposed algorithm.

Zhang, K., R. Statman and D. Shasha, On the editing distance between unordered labeled trees, Information Processing Letters 42 (1992) 133-139. This paper considers the problem of computing the editing distance between unordered, labeled trees. We give efficient polynomial-time algorithms for the case when one tree is a string or has a bounded number of leaves. By contrast, we show that the problem is NP-complete even for binary trees having a label alphabet of size two.

In this note, we introduce a graph invariant called the annihilation number and show that it is a sharp upper bound on the independence number. While the invariant does not distinguish between different graphs with the same degree sequence -since it is determined solely by the degrees -it still outperforms many well known upper bounds on independence number. The process which leads to the annihilation number is related to the process developed independently by Havel [17] and Hakimi [16] to determine whether or not there existed a simple graph with a given sequence of non-negative integers as its degree sequence.

We show that if the radius of a simple, connected graph equals its indepen-dence number, then the graph contains a Hamiltonian path. This result was conjectured by the computer program Graffiti.pc, using a new conjecture-generating strategy called Sophie. We also mention several other sufficient conditions for Hamiltonian paths that were conjectured by Graffiti.pc, but which are currently open, so far as we know.

The independent-domination number of a graph is the cardinality of a smallest set of mutually non-adjacent vertices which has the property that every vertex not in the set is adjacent to at least one that is. We present several conjectures made by the computer program Graffiti.pc about the independent-domination number of graphs, providing proofs and partial results for some of these conjectures.

The k-domination number γ k (G) of a simple, undirected graph G is the order of a smallest subset D of the vertices of G such that each vertex of G is either in D or adjacent to at least k vertices in D. In 2010, the conjecture-generating computer program, Graffiti.pc, was queried for upperbounds on the 2-domination number. In this paper we prove new upper bounds on the 2-domination number of a graph, some of which generalize to the k-domination number.

In this note, we introduce a graph invariant called the annihilation number and show that it is a sharp upper bound on the independence number. While the invariant does not distinguish between different graphs with the same degree sequence -since it is determined solely by the degrees -it still outperforms many well known upper bounds on independence number. The process which leads to the annihilation number is related to the process developed independently by Havel [17] and Hakimi [16] to determine whether or not there existed a simple graph with a given sequence of non-negative integers as its degree sequence.

The k-domination number of a graph is the cardinality of a smallest set of vertices such that every vertex not in the set is adjacent to at least k vertices of the set. We prove two bounds on the k-domination number of a graph, inspired by two conjectures of the computer program Graffiti.pc. In particular, we show that for any graph with minimum degree at least 2k -1, the k-domination number is at most the matching number.

An induced matching of a graph G is a matching having no two edges joined by an edge. An efficient edge dominating set of G is an induced matching M such that every other edge of G is adjacent to some edge in M. We relate maximum induced matchings and efficient edge dominating sets, showing that efficient edge dominating sets are maximum induced matchings, and that maximum induced matchings on regular graphs with efficient edge dominating sets are efficient edge dominating sets. A necessary condition for the existence of efficient edge dominating sets in terms of spectra of graphs is established. We also prove that, for arbitrary fixed p ≥ 3, deciding on the existence of efficient edge dominating sets on p-regular graphs is NP-complete.

Roman dominating function (or just RDF) on a graph G = (V, E) is a function f : V -→ {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. The weight of an RDF f is the value , of G is the minimum weight of an RDF on G. Several parameters related to the Roman dominating functions have been considered in the very recent years, for example, Roman bondage number, Roman reinforcement number, total Roman domination number, multiple Roman domination number, paired Roman domination number, and etc. We first establish several bounds for the Roman domination number of a graph under some given properties of the graph. We then determine the computational complexity of several Roman domination parameters, and show that the decision problem for these parameters are NP-complete even when restricted to bipartite graphs or chordal graphs. We also study Roman domination parameters in Random graphs.

hexamethyltriethylenetetramine; general formula C 3n N n H (7nϩ2) ] with four polycarboxylic ligands (malonate, citrate, 1,2,3-propanetricarboxylate, and 1,2,3,4-butanetetracarboxylate) has been studied potentiometrically in aqueous solution at 25ЊC. For all the systems, the species ALH r (r ϭ 1, 2 . . . n ϩ m Ϫ 1; n and m are the maximum degrees of protonation for the amine A and the carboxylic ligand L, respectively) are formed. The stability of these species is quite high (in particular, that of species with r ϭ n and r ϭ n ϩ 1) and is a linear function of the charges involved in the formation reaction. The effect of N-alkyl substitution is quite small: comparison with the analogous unsubstituted polyamines C (2nϪ2) N n H (5nϪ2) shows only a small decrease in stability. On average we have Ϫ⌬G o / n ϭ 5.9 kJ-mol Ϫ1 for N-alkyl-substituted amines and 6.6 kJ-mol Ϫ1 for unsubstituted ones (n denoting the number of possible salt bridges). Other linear chargestability relationships are considered.