A cycle-transversal of a graph G is a subset T ⊆ V (G) such that T ∩V (C) 6= ∅ for every cycle C of G. A clique cycle-transversal, or cct for short, is a cycle-transversal which is a clique. Recognizing graphs which admit a cct can be done in polynomial time; however, no structural characterization of such graphs is known. We characterize distancehereditary graphs admitting a cct in terms of fo...

The maximum independent set problem is known to be NP-hard for graphs in general, but is solvable in polynomial time for graphs in many special classes. It is also known that the problem is generally intractable from a parameterized point of view. A simple Ramsey argument implies the fixed parameter tractability of the maximum independent set problem in classes of graphs of bounded clique numbe...

We consider the problem of partitioning the set of vertices of a given unit disk graph (UDG) into a minimum number of cliques. The problem is NP-hard and various constant factor approximations are known, with the current best ratio of 3. Our main result is a polynomial time approximation scheme (PTAS) for this problem on UDG. In fact, we present a robust algorithm that given a graph G (not nece...

We obtain a sequence k,(G) Q k,(G) < *. . < k,(G) of lower bounds for the clique number (size of the largest clique) of a graph G of n vertices. The bounds involve the spectrum of the adjacency matrix of G. The bound k,(G) is explicit and improves earlier known theorems. The bound k,(G) is also explicit, and is shown to improve on the bound from Brooks’ theorem even for regular graphs. The boun...

Let C be a clique of a graph G. The capacity of C is defined to be (|V (G) \ C|+ |D|)/2, where D is the set of vertices in V (G) \C that have both a neighbour and a non-neighbour in C. We give a polynomial-time algorithm to find the minimum clique capacity in a graph G. This problem arose as an open question in a study [1] of packing vertex-disjoint induced three-vertex paths in a graph with no...

We provide a parameterized polynomial algorithm for the propositional model counting problem #SAT, the runtime of which is single-exponential in the rank-width of a formula. Previously, analogous algorithms have been known – e.g. [Fischer, Makowsky, and Ravve] – with a singleexponential dependency on the clique-width of a formula. Our algorithm thus presents an exponential runtime improvement (...

(1) It is equivalent to clique-width, a complexity measure introduced by Courcelle et al. [4], that generalises the well-known complexity measure tree-width introduced by Robertson and Seymour in their graph minors series. (2) It is algorithmically more interesting than clique-width because we can recognise in polynomial time graphs of rank-width at most k (for fixed k) (3) It shares with tree-...

The celebrated Erdős-Hajnal conjecture states that for every n-vertex undirected graph H there exists ε(H) > 0 such that every graph G that does not contain H as an induced subgraph contains a clique or an independent set of size at least n. A weaker version of the conjecture states that the polynomial-size clique/independent set phenomenon occurs if one excludes both H and its complement H. We...

Au algorithm is developed for findiug a close to optimal junction tree of a given graph Tile algorithm has a worst case co~nplexity O(ckn~) where a and c are coustants, the nmnber of vertices, and k is largest clique in a juuction tree of G in which this size is mini~nized. The algorithm guarantees that the logarittHn of the size of the state space of the heaviest clique in the junction tree pr...

The visibility graph V(P ) of a point set P ⊆ R has vertex set P , such that two points v, w ∈ P are adjacent whenever there is no other point in P on the line segment between v and w. We study the chromatic number of V(P ). We characterise the 2and 3-chromatic visibility graphs. It is an open problem whether the chromatic number of a visibility graph is bounded by its clique number. Our main r...