We study the fixed-parameter tractability, subexponential time computability, and approximability of the well-known NP-hard problems: INDEPENDENT SET, VERTEX COVER, and DOMINATING SET. We derive tight results and show that the computational complexity of these problems, with respect to the above complexity measures, is dependent on the genus of the underlying graph. For instance, we show that, ...

In a finite undirected graph G = (V,E), a vertex v ∈ V dominates itself and its neighbors in G. A vertex set D ⊆ V is an efficient dominating set (e.d. for short) of G if every v ∈ V is dominated in G 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 for P7-free graphs but solvable in linear time for P5-fr...

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 ...

We discuss matrix partition problems for graphs that admit a partition into k independent sets and ` cliques. We show that when k + ` 6 2, any matrix M has finitely many (k, `) minimal obstructions and hence all of these problems are polynomial time solvable. We provide upper bounds for the size of any (k, `) minimal obstruction when k = ` = 1 (split graphs), when k = 2, ` = 0 (bipartite graphs...

Finding maximum-cardinality matchings in undirected graphs is arguably one of the most central graph problems. For m-edge and n-vertex graphs, it is well-known to be solvable in O(m √ n) time; however, for several applications this running time is still too slow. Improving this worst-case upper bound resisted decades of research. In this paper we mainly focus on parameterizations of the input w...

DOMINATING SET remains NP -complete even when instances are restricted to bipartite graphs, however, in this case VERTEX COVER is solvable in polynomial time. Consequences to VECTOR DOMINATING SET as a generalization of both are discussed.

The maximum cardinality cut (MaxCut) problem remains NP-complete for co-bipartite graphs and
 split graphs. Based on modular decomposition, in [3] it is shown that MaxCut polynomial-time solvable are factorable to bounded treewidth However, this approach fails because the modules of a graph exactly twins and connected components. In work we extend result using concept bimodules bimodular d...

The thinness of a graph is width parameter that generalizes some properties interval graphs, which are exactly the graphs one. Graphs with at most two include, for example, bipartite convex graphs. Many NP-complete problems can be solved in polynomial time bounded thinness, given suitable representation graph. Proper defined analogously, generalizing proper and larger family known to polynomial...