Quasi-Tree Graphs with the Largest and the Second Largest Numbers of Maximum Independent Sets

Abstract

In a graph \(G = (V, E)\), an independent set is a subset \(I\) of \(V(G)\) such that no two vertices in \(I\) are adjacent. A maximum independent set is an independent set of maximum size. A connected graph (respectively, graph) \(G\) with vertex set \(V(G)\) is called a quasi-tree graph (respectively, quasi-forest graph), if there exists a vertex \(x \in V(G)\) such that \(G – x\) is a tree (respectively, forest). In this paper, we study the problem of determining the largest and the second largest numbers of maximum independent sets among all quasi-tree graphs and quasi-forest graphs. Extremal graphs achieving these values are also given.