Two characterizations of chain partitioned probe graphs

Abstract

Chain graphs are exactly bipartite graphs without induced 2K 2 (a graph with four vertices and two disjoint edges). A graph G = (V , E) with a given independent set S ⊆ V (a set of pairwise non-adjacent vertices) is said to be a chain partitioned probe graph if G can be extended to a chain graph by adding edges between certain vertices in S. In this note we give two characterizations for chain partitioned probe graphs. The first one describes chain partitioned probe graphs by six forbidden subgraphs. The second one characterizes these graphs via a certain "enhanced graph": G is a chain partitioned probe graph if and only if the enhanced graph G * is a chain graph. This is analogous to a result on interval (respectively, chordal, threshold, trivially perfect) partitioned probe graphs, and gives an O(m 2 )-time recognition algorithm for chain partitioned probe graphs.Given a graph class C, a graph G = (V , E) is called a C probe graph if there exists an independent set S ⊆ V and a set E ⊆ S 2 such that the graph G = (V , E ∪ E ) is in the class C, where S 2 stands for the set of all 2-element subsets of S. Probe graphs have been investigated for various graph classes; see Golumbic et al. (2009) and the literature given there for more information.In case of bipartite graphs without induced 2K 2 (a graph with four vertices and two disjoint edges), also called chain graphs by Yannakakis (1981) or difference graphs by Hammer et al. (1990), Golumbic et al. (2009) very recently found a characterization of chain probe graphs as follows; notice that chain probe graphs are necessarily bipartite. Theorem 1 (Golumbic et al. 2009) Let G be a bipartite graph. Then G is a chain probe graph if and only if it is {C 6 , P 7 , 3K 2 , H 1 , H 2 , H 3 }-free.

The chain graph sandwich problem

“…A characterization of chain probe graphs and an O(n 2 ) recognition algorithm was given in Golumbic et al (2009). The partitioned chain probe graph problem assumes that the independent set S is given and fixed in advance, and is a special case of the chain sandwich problem where E 2 \ E 1 = S × S. The complexity of the partitioned chain probe problem has also been shown to be polynomial in Van Bang (2010).…”

Good characterizations and linear time recognition for 2-probe block graphs

“…In case of probe graphs, i.e., k = 1 only few are known: In [1] it is shown that a graph is a partitioned probe threshold graph, respectively, a partitioned probe trivially perfect graph if and only if a certain enhanced graph is a threshold graph, respectively, a trivially perfect graph. In [14] it is shown that a graph is a partitioned chain graph if and only if a certain enhanced graph is a chain graph, and recently, [15] (cf. Theorem 1) proved that a graph is a partitioned block graph if and only if a certain enhanced graph is a block graph.…”