We show that the notoriously difficult problem of finding and reporting the smallest number of vertex-disjoint paths that cover the vertices of a graph can be solved timeand work-optimally for cographs. Our algorithm solves this problem in O(log n) time using n log n processors on the EREW-PRAM for an n-vertex cograph G represented by its cotree.

In this paper, we determine the minimal separators of P4-sparse graphs and establish bounds on their number. Specifically, we show that a P4-sparse graph G on n vertices and m edges has fewer than 2n/3 minimal separators of total description size at most 4m/3. The bound on the number of minimal separators is tight and is also tight for the class of cographs, a well known subclass of the P4-spar...

Transductions are a general formalism for expressing transformations of graphs (and more generally, relational structures) in logic. We prove that graph class C can be FO-transduced from bounded-height trees (that is, has bounded shrubdepth) if, and only one cannot FO-transduce the all paths. This establishes three remaining open questions posed by Blumensath Courcelle about MSO-transduction qu...

Jha, Pradhan, and Banerjee devised a linear algorithm to compute the secure domination number of cograph. Here it is shown that their Lemma 2, which crucial for computational complexity algorithm, incomplete. An accordingly modified lemma proved demonstrated remains linear.

We give a linear-time algorithm for computing the edge search number of cographs, thereby proving that this problem can be solved in polynomial time on this graph class. With our result, the knowledge on graph searching of cographs is now complete: node, mixed, and edge search numbers of cographs can all be computed efficiently. Furthermore, we are one step closer to computing the edge search n...

For a pair of natural numbers k,l, (k,l)-colouring graph G is partition the vertex set into (possibly empty) sets S1,S2,…,Sk, C1,C2,…,Cl such that each Si an independent and Cj induces clique in G. The problem, which NP-complete general, has been studied for special classes as chordal graphs, cographs line graphs. Let ?ˆ(G)=(?0(G),?1(G),…,??(G)?1(G)) ?ˆ(G)=(?0(G),?1(G),…,??(G)?1(G)) where ?l(G)...

In this paper, the space requirements for the OBDD representation of certain graph classes, specifically cographs, several types of graphs with few P4s, unit interval graphs, interval graphs and bipartite graphs are investigated. Upper and lower bounds are proven for all these graph classes and it is shown that in most (but not all) cases a representation of the graphs by OBDDs is advantageous ...