Forbidden graphs for classes of split-like graphs

Discrete Mathematics, 2005

A simple graph G has the generalized-neighbour-closed-co-neighbour property, or is a gncc graph, if for all vertices x of G, the subgraph, induced by the set of neighbours of x, is isomorphic to the subgraph, induced by the set of non-neighbours of x, or is isomorphic to its complement. If every vertex x satisfies the first condition (that is, the subgraphs, induced by its set of neighbours, and by its set of non-neighbours, are isomorphic), then the graph has the neighbour-closed-co-neighbour property, or is an ncc graph. In [A. Bonato, R. Nowakowski, Partitioning a graph into two isomorphic pieces, J. Graph Theory, 44 (2003) 1-14], the ncc graphs were characterized and a polynomial time algorithm was given for their recognition. In this paper we show that all gncc graphs are also ncc, that is, we prove that the two families of graphs, defined above, are identical. Finally, we present some of the properties of an interesting family of graphs, that is derived from the proof of the claim above, and we give a polynomial time algorithm to recognize such graphs.

Discrete Mathematics, 2009

Given integers j and k and a graph G, we consider partitions of the vertex set of G into j + k parts where j of these parts induce empty graphs and the remaining k induce cliques. If such a partition exists, we say G is a (j, k)-graph. For a fixed j and k we consider the maximum order n where every graph of order n is a (j, k)-graph. The split-chromatic number of G is the minimum j where G is a (j, j)-graph. Further, the cochromatic number is the minimum j+k where G is a (j, k)-graph. We examine some relations between cochromatic, split-chromatic and chromatic numbers. We also consider some computational questions related to chordal graphs and cographs.

2019

Given a finite undirected graph X, a vertex is 0-dismantlable if its open neighborhood is a cone and X is 0-dismantlable if it is reducible to a single vertex by successive deletions of 0-dismantlable vertices. By an iterative process, a vertex is pk ` 1q-dismantlable if its open neighborhood is kdismantlable and a graph is k-dismantlable if it is reducible to a single vertex by successive deletions of k-dismantlable vertices. Within the class of graphs whose clique complex is collapsible, the family of k-dismantlable graphs form a strict hierarchy in the sub-class of graphs whose clique complex is non-evasive. We introduce special graphs to study higher k-dismantlabilities. We point out how k-dismantlability is related to the derivability of graphs defined by Mazurkievicz and we get a new characterization of the class of closed graphs that he defined. By generalizing the notion of vertex transitivity, we replace k-dismantlability in its link with the evasiveness conjecture.

The Electronic Journal of Combinatorics

Can the vertices of an arbitrary graph $G$ be partitioned into $A \cup B$, so that $G[A]$ is a line-graph and $G[B]$ is a forest? Can $G$ be partitioned into a planar graph and a perfect graph? The NP-completeness of these problems are special cases of our result: if ${\cal P}$ and ${\cal Q}$ are additive induced-hereditary graph properties, then $({\cal P}, {\cal Q})$-colouring is NP-hard, with the sole exception of graph $2$-colouring (the case where both ${\cal P}$ and ${\cal Q}$ are the set ${\cal O}$ of finite edgeless graphs). Moreover, $({\cal P}, {\cal Q})$-colouring is NP-complete iff ${\cal P}$- and ${\cal Q}$-recognition are both in NP. This completes the proof of a conjecture of Kratochvíl and Schiermeyer, various authors having already settled many sub-cases.

Cornell University - arXiv, 2021

Stanisław Ulam asked whether there exists a universal countable planar graph (that is, a countable planar graph that contains every countable planar graph as a subgraph). János Pach (1981) answered this question in the negative. We strengthen this result by showing that every countable graph that contains all countable planar graphs must contain (i) an infinite complete graph as a minor, and (ii) a subdivision of the complete graph K t with multiplicity t, for every finite t. On the other hand, we construct a countable graph that contains all countable planar graphs and has several key properties such as linear colouring numbers, linear expansion, and every finite n-vertex subgraph has a balanced separator of size O(√ n). The graph is T 6 P ∞ , where T k is the universal treewidth-k countable graph (which we define explicitly), P ∞ is the 1-way infinite path, and denotes the strong product. More generally, for every positive integer t we construct a countable graph that contains every countable K t-minor-free graph and has the above key properties. Our final contribution is a construction of a countable graph that contains every countable K t-minor-free graph as an induced subgraph, has linear colouring numbers and linear expansion, and contains no subdivision of the countably infinite complete graph (implying (ii) above is best possible).

2005

We explore the properties of subgraphs (called Markovian subgraphs) of a decomposable graph under some condition. For a decomposable graph G and a collection γ of its Markovian subgraphs, we show that the set χ(G) of the intersections of all the neighboring cliques of G contains ∪g∈γχ(g). We also show that χ(G) = ∪g∈γχ(g) holds for a certain type of G which we call a maximal Markovian supergraph of γ.

Combinatorica, 2002

We prove a theorem about cutsets in partitionable graphs that generalizes earlier results on amalgams, 2-amalgams and homogeneous pairs.

2011

Some problems related to security in communication networks lead to consider a new type of connectivity in graphs, namely the confldential connectivity. In this paper we present a characterization of unbreakable graphs using the notion of weak decomposition and we give some applications of minimal unbreakable graphs. In fact, we showed that a graph G is confldentially connected if and only if it does not have a star cutset. We also showed that a minimal imperfect graph does not have a star cutset. We gave a constructive proof of the fact that every (fi;!)-partitionable graph is confldentially connected, for a superclass of minimal imperfect graphs.

Theoretical Computer Science, 2007

The substitution composition of two disjoint graphs G 1 and G 2 is obtained by first removing a vertex x from G 2 and then making every vertex in G 1 adjacent to all neighbours of x in G 2 . Let F be a family of graphs defined by a set Z of forbidden configurations. Giakoumakis [V. Giakoumakis, On the closure of graphs under substitution, Discrete Mathematics 177 (1997) 83-97] proved that F * , the closure under substitution of F, can be characterized by a set Z * of forbidden configurationsthe minimal prime extensions of Z. He also showed that Z * is not necessarily a finite set. Since substitution preserves many of the properties of the composed graphs, an important problem is the following: find necessary and sufficient conditions for the finiteness of Z * . Giakoumakis [V. Giakoumakis, On the closure of graphs under substitution, Discrete Mathematics 177 (1997) 83-97] presented a sufficient condition for the finiteness of Z * and a simple method for enumerating all its elements. Since then, many other researchers have studied various classes of graphs for which the substitution closure can be characterized by a finite set of forbidden configurations. The main contribution of this paper is to completely solve the above problem by characterizing all classes of graphs having a finite number of minimal prime extensions. We then go on to point out a simple way for generating an infinite number of minimal prime extensions for all the other classes of F * .

Combinatorica, 2007

A complete partition of a graph G is a partition of its vertex set in which any two distinct classes are connected by an edge. Let cp(G) denote the maximum number of classes in a complete partition of G. This measure was defined in 1969 by Gupta [G69], and is known to be NP-hard to compute for several classes of graphs. We obtain essentially tight lower and upper bounds on the approximability of this problem. We show that there is a randomized polynomial-time algorithm that given a graph G with n vertices, produces a complete partition of size Ω(cp(G)/ √ lg n). This algorithm can be derandomized. We show that the upper bound is essentially tight: there is a constant C > 1, such that if there is a randomized polynomial-time algorithm that for all large n, when given a graph G with n vertices produces a complete partition into at least C · cp(G)/ √ lg n classes, then NP ⊆ RTime(n O(lg lg n) ). The problem of finding a complete partition of a graph is thus the first natural problem whose approximation threshold has been determined to be of the form Θ((lg n) c ) for some constant c strictly between 0 and 1. * The work reported here is a merger of the results reported in [KRS05] and .