On Unique Independence Weighted Graphs

An independent set in a graph G is a set of vertices no two of which are joined by an edge. A vertex-weighted graph associates a weight with every vertex in the graph. A vertex-weighted graph G is called a unique independence vertex-weighted graph if it has a unique independent set with maximum sum of weights. Although, in this paper we observe that the problem of recognizing unique independence vertex-weighted graphs is NP-hard in general and therefore no efficient characterization can be expected in general; we give, however, some combinatorial characterizations of unique independence vertex-weighted graphs. This paper introduces a motivating application of this problem in the area of combinatorial auctions, as well. Introduction and preliminaries In this paper, we focus on graphs whose vertices have real weights and call such graphs for simplicity, just weighted graphs. Also, we study unique independent sets in finite vertex weighted graphs. For the definition of basic concepts and notations not given here one may refer to a textbook in graph theory, for example [G], and [I]. Let G = (V,E) be a simple undirected graph with the vertex set V = {1, 2, · · · , n}, the edge set E and a nonnegative weight w(i) associated with each vertex i ∈ V . The weight of S ⊆ V (G) is defined as w(S) = ∑ i∈S w(i). A subset I of V (G) is called an independent set (or a stable set) if the subgraph G[I] induced by I of G has no edges. A maximum weighted independent set, also called α-set, is an independent set of the largest weight in G. The weight of a maximum weighted independent set in G is denoted by α(G). A weighted graph G is a unique independence weighted graph, if G has a unique independent set with maximum sum of weights. Characterizing unique independence graphs and various generalizations of this concept has been a subject of research in graph theory literature. As a few examples, we refer the interested reader to [B], [D], and [C]. Also, some existing papers have focused on finding or even approximating the maximum independent set problem in weighted graphs. See [A], and [E] for more details. As we will observe in this paper, this is not coincidental: we show that the problem of recognizing unique maximum independence weighted graphs is NP-hard in general and therefore no 1991 Mathematics Subject Classification. Primary 05C69, 05C90; Secondary 68R10, 68Q15, 68Q17.