In a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be NP-complete even when restricted to bipartite graphs. It has been proved that Matching Cut is polynomially solvable for graphs of diameter two. In this paper, we show that, for any fixed integer d ≥ 4, Matching Cut is NP-comp...

While the famous Berge’s Strong Perfect Graph Conjecture (see [l] for details on perfect graphs) remains a major unsolved problem in Graph Theory, an alternative characterization of Perfect Graphs was conjectured in 1982 by Berge and the author [3]. This second conjecture asserts the existence of kernels for a certain type of orientations of perfect graphs. Here we prove a weaker form of the co...

In 2000 Alber et al. [SWAT 2000 ] obtained the first parameterized subexponential algorithm on undirected planar graphs by showing that k-DOMINATING SET is solvable in time 2O( √ k)nO(1), where n is the input size. This result triggered an extensive study of parameterized problems on planar and more general classes of sparse graphs and culminated in the creation of Bidimensionality Theory by De...

We consider the classical problems (Edge) Steiner Tree and Vertex after restricting input to some class of graphs characterized by a small set forbidden induced subgraphs. show dichotomy for former problem restricted (H1,H2)-free latter H-free graphs. find that there exists an infinite family H such is polynomial-time solvable graphs, whereas exist only two which this holds Edge (assuming P≠NP)...

Let G be a finite undirected graph. A vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d. in G, is known to be NP-complete even for very restricted graph classes such as P7-free chordal gr...

Let φ be Euler’s phi function. We prove that a vertex-transitive graph 0 of order n, with gcd(n, φ(n)) = 1, is isomorphic to a circulant graph of order n if and only if Aut(0) contains a transitive solvable subgroup. As a corollary, we prove that every vertex-transitive graph 0 of order n is isomorphic to a circulant graph of order n if and only if for every such 0, Aut(0) contains a transitive...

A graph is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest k for which such a coloring exists is known as the equitable chromatic number of G and it is denoted by χ=(G). In this paper the problem of determinig χ= for coronas of cubic graphs is studied. Although the prob...

A graph class is hereditary if it closed under vertex deletion. We give examples of NP-hard, PSPACE-complete and NEXPTIME-complete problems that become constant-time solvable for every not equal to the all graphs.

We continue research into a well-studied family of problems that ask whether the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a graph from some specified graph class G. We let G be the class of k-degenerate graphs. This problem is known to be polynomial-time solvable if k = 0 (bipartite graphs) and NPcomplete if k = 1 (near-bipartite grap...

A disconnected cut of a connected graph is a vertex cut that itself also induces a disconnected subgraph. The decision problem whether a graph has a disconnected cut is called Disconnected Cut. This problem is closely related to several homomorphism and contraction problems, and fits in an extensive line of research on vertex cuts with additional properties. It is known that Disconnected Cut is...