Whether the clique-width of graphs in a certain class is bounded or not is an important question from an algorithmic point of view, since many problems that are NP-hard in general admit polynomial-time solutions when restricted to graphs of bounded clique-width. Over the last few years many classes of graphs have been shown to have bounded clique-width. For many, this parameter has been proved ...

Clique graphs of chordal and path graphs are characterized. A special class of graphs named expanded trees is discussed. It consists of a subclass of disk-Helly graphs. It is shown that the clique graph of every chordal (hence path) graph is an expanded tree. In addition, every expanded tree is the clique graph of some path (hence chordal) graph. Diierent characterizations of expanded trees are...

Many NP-complete graph problems are polynomial-time solvable on classes of bounded clique-width. Several these a hereditary class G ${\mathscr{G}}$ if they so the atoms (graphs with no clique cut-set) . Hence, we initiate systematic study into boundedness clique-width classes. A $G$ is H $H$ -free not an induced subgraph , and it ( 1 2 ) $({H}_{1},{H}_{2})$ both ${H}_{1}$ ${H}_{2}$ -free. graph...

It is well known that the coefficients of matching polynomial are unimodal. Unimodality (or their absolute values) other graph polynomials has been studied as well. One way to prove unimodality real-rootedness. Recently I. Beaton and J. Brown (2020) proved for almost all graphs domination form a unimodal sequence, C. Barton, D. Pike forest (aka acyclic polynomial) real-rooted iff G forest. Let ...

The clique-width is a measure of complexity of decomposing graphs into certain tree-like structures. The class of graphs with bounded clique-width contains bounded tree-width graphs. While there are many results on the graph isomorphism problem for bounded treewidth graphs, very little is known about isomorphism of bounded cliquewidth graphs. We give the first polynomial-time graph isomorphism ...

The k-fold clique transversal problem is to locate a minimum set of vertices of a graph such that every maximal clique has at least k elements of. The maximum h-colourable subgraph problem is to nd a maximum subgraph of a graph which is h-colourable. We show that the k-fold clique transversal problem and the maximum h-colourable subgraph problem are polynomially solvable on balanced graphs. We ...

Bayesian networks (BNs) are used to represent and efficiently compute with multi-variate probability distributions in a wide range of disciplines. One of the main approaches to perform computation in BNs is clique tree clustering and propagation. In this approach, BN computation consists of propagation in a clique tree compiled from a Bayesian network. There is a lack of understanding of how cl...

Similar to the tree-width (twd), the clique-width (cwd) is an invariant of graphs. A well known relationship between tree-width and clique-width is that cwd(G) ≤ 3 · 2. It is also known that treewidth of Cactus graphs is 2, therefore the clique-width for those graphs is smaller or equal than 6. In this paper, it is shown that the clique-width of Cactus graphs is smaller or equal to 4 and we pre...

Biró, Hujter, and Tuza introduced the concept of H-graphs (1992), intersection graphs of connected subgraphs of a subdivision of a graph H. They naturally generalize many important classes of graphs, e.g., interval graphs and circular-arc graphs. We continue the study of these graph classes by considering coloring, clique, and isomorphism problems on H-graphs. We show that for any fixed H conta...

In an attempt to find a polynomial-time algorithm for the edgeclique cover problem on cographs we tried to prove that the edge-clique graphs of cographs have bounded rankwidth. However, this is not the case. In this note we show that the edge-clique graphs of cocktail party graphs have unbounded rankwidth.