Strongly Chordal and Chordal Bipartite Graphs Are Sandwich Monotone

Abstract

A graph class is sandwich monotone if, for every pair of its graphs G 1 = (V ,E 1 ) and G 2 = (V ,E 2 ) with E 1 *** E 2 , there is an ordering e 1 , ..., e k of the edges in E 2 *** E 1 such that G = (V , E 1 *** {e 1 , ..., e i }) belongs to the class for every i between 1 and k . In this paper we show that strongly chordal graphs and chordal bipartite graphs are sandwich monotone, answering an open question by Bakonyi and Bono from 1997. So far, very few classes have been proved to be sandwich monotone, and the most famous of these are chordal graphs. Sandwich monotonicity of a graph class implies that minimal completions of arbitrary graphs into that class can be recognized and computed in polynomial time. For minimal completions into strongly chordal or chordal bipartite graphs no polynomial-time algorithm has been known. With our results such algorithms follow for both classes. In addition, from our results it follows that all strongly chordal graphs and all chordal bipartite graphs with edge constraints can be listed efficiently.