On Unique Independence Weighted Graphs

TL;DR

This paper studies the properties of unique independence weighted graphs, explores their recognition complexity, and provides combinatorial characterizations, with applications in combinatorial auctions.

Contribution

It introduces combinatorial characterizations of unique independence weighted graphs and discusses their recognition complexity, highlighting applications in auctions.

Findings

Recognizing unique independence weighted graphs is NP-hard.

Provides combinatorial characterizations of such graphs.

Highlights application in combinatorial auctions.

Abstract

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.