The chromatic number of a graph G, denoted by χ(G), is the minimum number of colors such that G can be colored with these colors in such a way that no two adjacent vertices have the same color. A clique in a graph is a set of mutually adjacent vertices. The maximum size of a clique in a graph G is called the clique number of G. The Turán graph Tn(k) is a complete k-partite graph whose partition...

A connection tree of a graph G for terminal set W is subgraph T such that leaves( ) ⊆ V(T) . non-terminal vertex called linker if its degree in exactly 2, and it router at least 3. The erminal problem (TCP) asks whether admits with most ℓ linkers r routers, while the S teiner k vertices. We prove that, ≥ 1 fixed, then TCP polynomial-time solvable when restricted to split graphs. This result sep...

Subtree filament graphs are the intersection graphs of subtree filaments in a tree. This class of graphs contains subtree overlap graphs, interval filament graphs, chordal graphs, circle graphs, circular-arc graphs, cocomparability graphs, and polygon-circle graphs. In this paper we show that, for circle graphs, the clique cover problem is NP-complete and the h-clique cover problem for fixed h ...

For a fixed k, an oriented graph is k-transitive if it is acyclic and for every directed path p=u1→u2→...→uk+2 with k+2 vertices, p induces a clique if each of the two subpaths u1→u2→...→uk+1 and u2→...→uk+2 induces a clique. We describe an algorithm to find a maximum weight clique in a k-transitive graph. Consider a hereditary family G of graphs. A graph H(V,E) is called G-k-mixed if its edge ...

Meta-kernelization theorems are general results that provide polynomial kernels for large classes of parameterized problems. The known meta-kernelization theorems, in particular the results of Bodlaender et al. (FOCS’09) and of Fomin et al. (FOCS’10), apply to optimization problems parameterized by solution size. We present meta-kernelization theorems that use a structural parameters of the inp...

Let G = (V,E) be a (finite, simple, undirected) graph with vertex set V and edge set E. A clique in G is a set of vertices, each of which is adjacent to every other vertex in this set. The maximum clique problem is to find a largest such set in a given graph; this is NP-hard [GJ90]. We denote the size of a maximum clique by ω. A colouring of G is an assignment of vertices to colours such that n...

Many problems can be formulated as graphs where a graph consists of a set of vertices and a set of edges, in which the vertices stand for objects in question and the edges stand for some relations among the objects. A clique is a subgraph in which all pairs of vertices are mutually adjacent. Thus, a maximum clique stands for a maximum collection of objects which are mutually related in some spe...

List homomorphisms generalize list colourings in the following way: Given graphs G; H , and lists L(v) V (H); v 2 V (G), a list homomorphism of G to H with respect to the lists L is a mapping f : V (G) ! V (H) such that uv 2 E (G) implies f (u)f(v) 2 E (H), and f (v) 2 L(v) for all v 2 V (G). The list homomorphism problem for a xed graph H asks whether or not an input graph G together with list...

We study algorithms for ]SAT and its generalized version ]GENSAT, the problem of computing the number of satisfying assignments of a set of propositional clauses . For this purpose we consider the clauses given by their incidence graph, a signed bipartite graph SI( ), and its derived graphs I( ) and P ( ). It is well known, that, given a graph of tree-width k, a k-tree decomposition can be foun...