Two sets are non-crossing if they are disjoint or one contains the other. The noncrossing graph NCn is the graph whose vertex set is the set of nonempty subsets of [n] = {1, . . . , n} with an edge between any two non-crossing sets. Various facts, some new and some already known, concerning the chromatic number, fractional chromatic number, independence number, clique number and clique cover nu...

A hole is a chordless cycle with at least four vertices. A pan is a graph which consists of a hole and a single vertex with precisely one neighbor on the hole. An even hole is a hole with an even number of vertices. We prove that a (pan, even hole)-free graph can be decomposed by clique cutsets into essentially unit circular-arc graphs. This structure theorem is the basis of our O(nm)-time cert...

This paper studies the maximum-clique independence problem and some variations of clique transversal such as {k}-clique, maximum-clique, minus clique, signed k-fold problems from algorithmic aspects for k-trees, suns, planar graphs, doubly chordal perfect total split line dually graphs. We give equations to compute numbers show that {k}-clique is polynomial-time solvable graphs whose equal thei...

In this paper we address the problem of finding all maximal cliques in a subclass of circular-arc graphs. We consider only circular-arc graphs that are generated with an arc model where no three arcs cover the whole circle. We prove that the number of maximal cliques is bounded by the number of vertices and that for each maximal clique it exists a point in the circle which is covered by all arc...

Assume that G = G(V, E) is an undirected graph with vertex set V and edge set E. A clique of G is a complete subgraph. An edge clique-covering is a family of cliques of G which cover all edges of G. The edge clique-cover number, Be( G), is the minimum number of cliques in an edge clique-cover of G. For results and applications of the edge clique-cover number see [1-4]. Observe that Be(G) does n...

A main result of combinatorial optimization is that clique and chromatic number of a perfect graph are computable in polynomial time (Grötschel, Lovász and Schrijver 1981). Circular-perfect graphs form a well-studied superclass of perfect graphs. We extend the above result for perfect graphs by showing that clique and chromatic number of a circularperfect graph are computable in polynomial time...

A class of graphs G is χ-bounded if the chromatic number of graphs in G is bounded by a function of the clique number. We show that if a class G is χ-bounded, then every class of graphs admitting a decomposition along cuts of small rank to graphs from G is χ-bounded. As a corollary, we obtain that every class of graphs with bounded rank-width (or equivalently, clique-width) is χ-bounded.

Graphs with circular symmetry, called webs, are relevant for describing the stable set polytopes of two larger graph classes, quasi-line graphs [6,10] and claw-free graphs [5,6]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a longstanding problem [7]. However, even the problem of finding all facets of stable set polytopes of webs is open. So far, it i...

In this paper we investigate a variation of the graph coloring game, as studied in [2]. In the original coloring game, two players, Alice and Bob, alternate coloring vertices on a graph with legal colors from a fixed color set, where a color α is legal for a vertex if said vertex has no neighbors colored α. Other variations of the game change this definition of a legal color. For a fixed color ...

Abstract A clique coloring of a graph is an assignment colors to its vertices such that no maximal monochromatic. We initiate the study structural parameterizations Clique Coloring problem which asks whether given has with q colors. For fixed $$q \ge 2$$ q ≥ 2 </mm...