Given two graphs H1 and H2, a graph G is (H1,H2)-free if it contains no subgraph isomorphic to H1 or H2. We continue a recent study into the clique-width of (H1,H2)-free graphs and present three new classes of (H1, H2)-free graphs that have bounded clique-width. We also show the implications of our results for the computational complexity of the Colouring problem restricted to (H1,H2)-free grap...

In this paper we deal with a Turán-type problem: given a positive integer n and a forbidden graph H , how many edges can there be in a graph on n vertices without a subgraph H? How does a graph look like if it has this extremal edge number? The forbidden graph in this article is a clique-path: a path of length k where each edge is extended to an r-clique, r ≥ 3. We determine both the extremal n...

A finite metric is h − embeddable if it can be embedded isometrically in the N -cube (hypercube) for some N . It is known that the problem of testing whether a metric is h − embeddable is NP-Complete, even if the distances are restricted to the set {2, 4, 6 }. Here we study the problem where the distances are restricted to the set {1, 2, 3 } and give a polynomial time algorithm and forbidden su...

We study the linear clique-width of graphs that are obtained from paths by disjoint union and adding true twins. We show that these graphs have linear clique-width at most 4, and we give a complete characterisation of their linear clique-width by forbidden induced subgraphs. As a consequence, we obtain a linear-time algorithm for computing the linear clique-width of the considered graphs. Our r...

A connected edge-colored graph G is rainbow-connected if any two distinct vertices of G are connected by a path whose edges have pairwise distinct colors; the rainbow connection number rc(G) of G is the minimum number of colors such that G is rainbow-connected. We consider families F of connected graphs for which there is a constant kF such that, for every connected F-free graph G, rc(G) ≤ diam...

Abstract: The intersection graph of a set of chords on a circle is called a circle graph. This class of graphs admits some interesting subclasses, such as: Helly circle graphs, clique-Helly circle graphs, unit circle graphs and proper circular-arc graphs. In this paper, we prove some inclusion relations among these subclasses. A necessary condition for a graph being a Helly circle graph is show...

In this paper, we compare the offline versions of three Ramsey-type oneplayer games that have been studied in an online setting in previous work: the online Ramsey game, the balanced online Ramsey game, and the Achlioptas game. The goal in all games is to color the edges of the random graph Gn,m according to certain rules without creating a monochromatic copy of some fixed forbidden graph H. Wh...

Zhu [15] introduced circular-perfect graphs as a superclass of the well-known perfect graphs and as an important χ-bound class of graphs with the smallest non-trivial χ-binding function χ(G) ≤ ω(G)+1. Perfect graphs have been recently characterized as those graphs without odd holes and odd antiholes as induced subgraphs [4]; in particular, perfect graphs are closed under complementation [7]. In...

We extend the notion of a minor from matroids to simplicial complexes. We show that the class of matroids, as well as the class of independence complexes, is characterized by a single forbidden minor. Inspired by a recent result of Aharoni and Berger, we investigate possible ways to extend the matroid intersection theorem to simplicial complexes.