Independent Domination: Reductions from Circular- and Triad-Convex Bipartite Graphs to Convex Bipartite Graphs

Abstract

An independent dominating set in a graph is a subset of vertices, such that no edge has both ends in the subset, and each vertex either itself is in the subset or has a neighbor in the subset. In a convex bipartite (circular convex bipartite, triad convex bipartite, respectively) graph, there is a linear ordering (a circular ordering, a triad, respectively) defined on one class of vertices, such that for every vertex in the other class, the neighborhood of this vertex is an interval (a circular arc, a subtree, respectively), where a triad is three paths with a common end. The problem of finding a minimum independent dominating set, called independent domination, is known \(\mathcal{NP}\)-complete for bipartite graphs and tractable for convex bipartite graphs. In this paper, we make polynomial time reductions for independent domination from triad- and circular-convex bipartite graphs to convex bipartite graphs.