We present a parallel algorithm for recognizing cographs and constructing their cotrees. The algorithm takes O(log 2 n) time with O(n + m) processors on a CRCW PRAM, where n and m are the number of vertices and edges of the graph. Using cotree representation, we obtain parallel algorithms for solving the maximum matching and the permutation representation problems for cographs using O(logn) tim...

In this paper,we study a variant of thepath cover problem, namely, the k -fixed-endpoint path cover problem. Given a graph G and a subset T of k vertices of V (G), a k fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint paths P that covers the vertices of G such that the k vertices ofT are all endpoints of the paths inP . The k -fixed-endpoint path cover problem is to f...

We show that deciding whether a given graph G of size m has a unique perfect matching as well as finding that matching, if it exists, can be done in time O(m) if G is either a cograph, or a split graph, or an interval graph, or claw-free. Furthermore, we provide a constructive characterization of the claw-free graphs with a unique perfect matching.

Densest k-Subgraph is the problem to find a vertex subset S of size k such that number edges in subgraph induced by maximized. In this paper, we show fixed parameter tractable when parameterized neighborhood diversity, block deletion number, distance-hereditary and cograph respectively. Furthermore, give 2-approximation \(2^{{{\texttt{tc}}(G)}/2}n^{O(1)}\)-time algorithm where \({{\texttt{tc}}(...

TheGrundynumber of a graph is themaximal number of colors attained by a first-fit coloring of the graph. The class of Cameron graphs is the Seidel switching class of cographs. In this paper we show that the Grundy number is computable in polynomial time for Cameron graphs.

We examine powers of Hamiltonian paths and cycles as well as Hamiltonian (power) completion problems in several highly structured graph classes. For threshold graphs we give efficient algorithms as well as sufficient and minimax toughness like conditions. For arborescent comparability graphs we have similar results but also show that for one type of completion problem an ‘obvious’ minimax condi...

In this paper we consider module-composed graphs, i.e. graphs which can be defined by a sequence of single vertex insertions v1, . . . , vn, such that the neighbourhood of vertex vi, 2 ≤ i ≤ n, forms a module (a homogenous set) of the graph defined by vertices v1, . . . , vi−1. Module composed graphs are HHDS-free and thus homogeneous orderable, weakly chordal, and perfect. Every bipartite dist...

For a graph G and a set of graphs H, we say that G is H-free if no induced subgraph of G is isomorphic to a member of H. Given an integer P > 0, a graph G, and a set of graphs F , we say that G admits an (F , P )-partition if the vertex set of G can be partitioned into P subsets X1, . . . ,XP , so that for every i ∈ {1, . . . , P}, either |Xi| = 1, or the subgraph of G induced by Xi is {F}-free...

We give a linear-time algorithm for computing the edge search number of cographs, thereby resolving the computational complexity of edge searching on this graph class. To achieve this we give a characterization of the edge search number of the join of two graphs. With our result, the knowledge on graph searching of cographs is now complete: node, mixed, and edge search numbers of cographs can a...

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...