A cograph completion of an arbitrary graph G is a cograph supergraph of G on the same vertex set. Such a completion is called minimal if the set of edges added to G is inclusion minimal. In this paper we characterize minimal cograph completions, and we give the following linear-time algorithms: one for extracting a minimal cograph completion from any given cograph completion of G, and one for d...

A cograph is a graph that contains no path on four vertices as an induced subgraph. A cograph k-partition of a graph G = (V,E) is a vertex-partition of G into k sets V1, . . . , Vk ⊂ V so that the graph induced by Vi is a cograph for 1 ≤ i ≤ k. Gimbel and Nešetřil [5] studied the complexity aspects of the cograph k-partitions and raised the following questions: Does there exist a triangle-free ...

A cograph completion of an arbitrary graph G is a cograph supergraph of G on the same vertex set. Such a completion is called minimal if the set of edges added to G is inclusion minimal. In this paper we present two results on minimal cograph completions. The first is a a characterization that allows us to check in linear time whether a given cograph completion is minimal. The second result is ...

A cograph completion of an arbitrary graph G is a cograph supergraph of G on the same vertex set. Such a completion is called minimal if the set of edges added to G is inclusion minimal. In this paper we present two results on minimal cograph completions. The first is a a characterization that allows us to check in linear time whether a given cograph completion is minimal. The second result is ...

A graph is called a cograph contraction if it is obtained from a cograph (a graph with no induced path on four vertices) by contracting some pairwise disjoint independent sets and then making the ‘‘contracted’’ vertices pairwise adjacent. Cograph contractions are perfect and generalize cographs and split graphs. This article gives a good characterization of cograph contractions, solving a probl...

Let G be a graph and letN1, . . . ,Nk be k independent sets in G. The graph G is a k-probe cograph if G can be embedded into a cograph by adding edges between vertices that are contained in the same independent set. We show that there exists an O(k ·n) algorithm to check if a graph G is a k-probe cograph.

A cograph is a graph which can be generated by disjoint union and complement operations on graphs, starting with a single vertex graph. Toward effective data mining for graph structured data, we introduce a graph pattern expression, called a cograph pattern, based on cographs. We show that the class of cograph pattern languages is polynomial time inductively inferable from positive data.

A cograph is a graph without induced P4 (where P4 denotes an induced path with 4 vertices). A graph G is (k, `) if its vertex set can be partitioned into at most k independent sets and ` cliques. Threshold graphs are cographs-(1, 1). Cographs-(2, 1) are a generalization of threshold graphs and, as threshold graphs, they can be recognized in linear time. GRAPH SANDWICH PROBLEMS FOR PROPERTY Π (Π...

We present a dynamic programming algorithm for optimally solving the Cograph Editing problem on an n-vertex graph that runs in O(3n) time and uses O(2) space. In this problem, we are given a graph G = (V,E) and the task is to find a smallest possible set F ⊆ V × V of vertex pairs such that (V,E4F ) is a cograph (or P4-free graph), where 4 represents the symmetric difference operator. We also de...

Cographs are those graphs without induced path on four vertices. A graph G is a probe cograph if its vertex set can be partitioned into two sets, N (non-probes) and P (probes) where N is independent and G can be extended to a cograph by adding edges between certain non-probes. A partitioned probe cograph is a probe cograph with a given partition in N and P . We characterise probe cographs in se...