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Recently, D. Corneil found a simple 3-sweep lexicographic breadth first search (LexBFS) algorithm for the recognition of proper interval graphs. We point out how to modify Corneil's algorithm to make it a certifying algorithm, and then describe a similar certifying 3-sweep LexBFS algorithm for the recognition of proper interval bigraphs. It follows from an earlier paper that the class of proper interval bigraphs is equal to the better known class of bipartite permutation graphs, and so we have a certifying algorithm for that class as well. All our algorithms run in time O(m+n), including the certification phase. The certificates of representability (the intervals) can be authenticated in time O(m + n). The certificates of nonrepresentability (the forbidden subgraphs) can be authenticated in time O(n). [ABSTRACT FROM AUTHOR]

A pair of vertices of a graph is called a dominating pair if the vertex set of every path between these two vertices is a dominating set of the graph. A graph is a weak dominating pair graph if it has a dominating pair. Further, a graph is called a dominating pair graph if each of its connected induced subgraphs is a weak dominating pair graph. Dominating pair graphs form a class of graphs containing interval, permutation, cocomparability, and asteroidal triple-free graphs. Our purpose is to study the structural properties of dominating pair graphs. Our main results are a polar theorem for the dominating pairs in weak dominating pair graphs and an existence theorem for minimum cardinality connected dominating sets that induce a simple path in connected dominating pair graphs of diameter not equal to three. Furthermore, we present a forbidden induced subgraph characterization of chordal dominating pair graphs. [ABSTRACT FROM AUTHOR]