Recently lexicographic breadth first search (LexBFS) has been shown to be a very powerful tool for the development of linear time, easily implementable recognition algorithms for various families of graphs. In this paper, we add to this work by producing a simple two LexBFS sweep algorithm to recognize the family of cographs. This algorithm extends to other related graph families such as P4-red...

In this paper, we describe a new simple linear time algorithm to recognize cographs. Cographs are exactly the P4-free graphs (where P4 denotes the path with 4 vertices). The recognition process works in two steps. First, we use partition refinement techniques to produce a factorizing permutation, i.e., an ordering of the vertices in which the strong modules appear consecutively. Then a very sim...

Generating Sparse 2-Spanners p. 73 Low-Diameter Graph Decomposition is in NC p. 83 Parallel Algorithm for Cograph Recognition with Applications p. 94 Parallel Algorithms for All Minimum Link Paths and Link Center Problems p. 106 Optimal Multi-Packet Routing on the Torus p. 118 Parallel Algorithms for Priority Queue Operations p. 130 Heap Construction in the Parallel Comparison Tree Model p. 140...

In this paper, we examine the controllability of Laplacian dynamic networks on cographs. Cographs appear in modeling a wide range and include as special instances, threshold graphs. work, present necessary sufficient conditions for cographs, provide an efficient method selecting minimal set input nodes from which network is controllable. particular, define sibling partition cograph show that co...

Cographs are a well-known class of graphs arising in a wide spectrum of practical applications. In this note, we show that the connected components of a cograph G can be optimally found in O(log log log (G)) time using O( (n+m) log log log (G) ) processors on a common CRCW PRAM, or in O(log (G)) time using O( (n+m) log (G) ) processors on an EREW PRAM, where (G) is the maximum degree of G, and ...

In this paper we show that the contiguity and linearity of cographs on n vertices are both O(log n). Moreover, we show that this bound is tight for contiguity as there exists a family of cographs on n vertices whose contiguity is Ω(log n). We also provide an Ω(log n/ log log n) lower bound on the maximum linearity of cographs on n vertices. As a by-product of our proofs, we obtain a min-max the...

Two independent sets of a graph are adjacent if they differ on exactly one vertex (i.e. we can transform one into the other by adding or deleting a vertex). Let k be an integer. We consider the reconfiguration graph TARk(G) on the set of independent sets of size at least k in a graph G, with the above notion of adjacency. Here we provide a cubic-time algorithm to decide whether TARk(G) is conne...

Matrix partition problems generalize a number of natural graph partition problems, and have been studied for several standard graph classes. We prove that each matrix partition problem has only finitely many minimal obstructions for split graphs. Previously such a result was only known for the class of cographs. (In particular, there are matrix partition problems which have infinitely many mini...

A feedback vertex set in a graph is a set of vertices whose removal leaves the remaining graph acyclic. Given the vast number of published results concerning feedback vertex sets, it is surprising that the related combinatorics appears to be so poorly understood. The maximum number of minimal feedback vertex sets in a graph on n vertices is known to be at most 1.864. However, no examples of gra...

A graph is polar if the vertex set can be partitioned into A and B in such a way that A induces a complete multipartite graph and B induces a disjoint union of cliques (i.e., the complement of a complete multipartite graph). Polar graphs naturally generalize several classes of graphs such as bipartite graphs, cobipartite graphs and split graphs. Recognizing polar graphs is an NP-complete proble...