On Intersection Representations and Clique Partitions of Graphs

Journal of Graph Theory

Throughout, G = (V , E) refers to a simple loopless graph and n denotes the number of vertices in G when the graph in question is clear. Standard graph theory notation is used throughout, but we make the following definitions explicit. We denote by G [X ] the subgraph of G induced by X ⊆ V (G). A clique is a set Q ⊆ V (G) such that G [Q ] is a complete graph. The edge clique cover number of a graph G , denoted ecc(G), is the minimum number of complete subgraphs of G whose union contains every edge of G. This parameter, also known as the intersection number of a graph, was introduced by Erdős, Goodman, and Posa [1]. In addition to being an interesting parameter in its own right, it is also related to a parameter known as the competition number of a graph, introduced by Cohen [2] in the study of food webs.

Bull. ICA, 2020

If X is any nonempty set on n ≥ 2 elements we define the set graph Gn to be the graph whose vertices are the 2 − 2 proper subsets of X with two vertices adjacent if and only if their underlying sets are disjoint. We discuss some properties of Gn. In particular we find its clique partition number and its product dimension. We also give bounds for its representation number. We use standard graph theory terminology as given in [13]. A family of subsets S1, S2, . . . of a set S gives a graph in a natural way if we use these sets as vertices and let SiSj for i 6= j be an edge if and only if the corresponding subsets have a nonempty intersection. In [12], Marczewski has established the converse, i.e. for any graph G there is a set S, such that a family of its subsets defines G according to the above description. Erdős, Goodman and Posa in [1] have remarked that one may replace the idea of a nonempty intersection with disjointness of the subsets since the same would then imply Marczewski’s...

Applied Mathematical Sciences, 2014

Let G be a graph. A clique of a graph G is a nonempty subset S of V (G) such that S induces a complete graph. A family of cliques of G is a clique cover of G if for every u ∈ V (G) there exists S ∈ such that u ∈ S. The clique covering number of G, denoted by cc(G), is given by cc(G) = min{| | : is a clique cover G}. For a graph G, the family of singleton subsets of V (G) is a clique cover of G. Hence, the order of a graph is an upper bound for its clique covering number. The clique covering number of a graph is equal to its order if and only if it is an empty graph. The clique covering number of a graph is equal to its size if and only if it is a star.

Discrete Applied Mathematics, 2013

Let G be a graph on n vertices. We call a subset A of the vertex set

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A family $\mathcal{S}$ of nonempty sets $\{S_1,\ldots,S_n\}$ is a set representation of $G$ if there exists a one-to-one correspondence between the vertices $v_1, \ldots, v_n$ in $V(G)$ and the sets in $\mathcal{S}$ such that $v_iv_j \in E(G)$ if and only if $S_i\cap S_j\neq \es$. A set representation $\mathcal{S}$ is a distinct (respectively, antichain, uniform and simple) set representation if any two sets $S_i$ and $S_j$ in $\mathcal{S}$ have the property $S_i\neq S_j$ (respectively, $S_i\nsubseteq S_j$, $|S_i|=|S_j|$ and $|S_i\cap S_j|\leqslant 1$). Let $U(\mathcal{S})=\bigcup_{i=1}^n S_i$. Two set representations $\mathcal{S}$ and $\mathcal{S}'$ are isomorphic if $\mathcal{S}'$ can be obtained from $\mathcal{S}$ by a bijection from $U(\mathcal{S})$ to $U(\mathcal{S}')$. Let $F$ denote a class of set representations of a graph $G$. The type of $F$ is the number of equivalence classes under the isomorphism...

Indagationes Mathematicae (Proceedings), 1977

Discrete Applied Mathematics, 2015

Clique-width of graphs is defined algebraically through operations on graphs with vertex labels. We characterise the clique-width in a combinatorial way by means of partitions of the vertex set, using trees of nested partitions where partitions are ordered bottom-up by refinement. We show that the correspondences in both directions, between combinatorial partition trees and algebraic terms, preserve the tree structures and that they are computable in polynomial time. We apply our characterisation to linear clique-width. And we relate our characterisation to a clique-width characterisation by Heule and Szeider that is used to reduce the clique-width decision problem to a satisfiability problem.

Asia Pacific Journal of Mathematics, 2021

A. In this paper we study the generalized complement of the graph G m,n = (V, E) for some values of m, n. We study the generalized complement of G m,n graphs with respect to the equal degree partition. The 2−complement of G m,n graphs are also determined for m = 2, n is even or odd. In particular, for some values of m, n ∈ N, we studied the complement of G m,n graphs with respect to the equal degree partition and the 2−complement of G m,n graphs. We determine the partitions P k , k ∈ N of the vertex set V such that the generalized complement of G m,n graph is a path graph and a comb graph.

Discrete Mathematics, 1991

Wallis, W.D. and J. Wu, On clique partitions of split graphs, Discrete Mathematics 92 (1991) 427-429. Split graphs are graphs formed by taking a complete graph and an empty graph disjoint from it and some or all of the possible edges joining the two. We prove that the problem of deciding the clique partition number is NP-complete, even when restricted to the class of split graphs.

A k−clique covering of a simple graph G, is an edge covering of G by its cliques such that each vertex is contained in at most k cliques. The smallest k for which G admits a k−clique covering is called local clique cover number of G and is denoted by lcc(G). Local clique cover number can be viewed as the local counterpart of the clique cover number which is equal to the minimum total number of cliques covering all edges. In this paper, several aspects of the problem are studied and its relationships to other well-known problems are discussed. Moreover, the local clique cover number of claw-free graphs and its subclasses are notably investigated. In particular, it is proved that local clique cover number of every claw-free graph is at most c∆/ log ∆, where ∆ is the maximum degree of the graph and c is a universal constant. It is also shown that the bound is tight, up to a constant factor. Furthermore, it is established that local clique number of the linear interval graphs is bounded by log ∆ + 1/2 log log ∆ + O(1). Finally, as a by-product, a new Bollobás-type inequality is obtained for the intersecting pairs of set systems. arXiv:1210.6965v1 [math.CO]