On the sizes of generalized cactus graphs

Abstract

A cactus is a connected graph in which each edge is contained in at most one cycle. We generalize the concept of cactus graphs, i.e., a k-cactus is a connected graph in which each edge is contained in at most k cycles where k≥1. It is well known that every cactus with n vertices has at most ⌊32(n−1)⌋ edges. Inspired by it, we attempt to establish analogous upper bounds for general k-cactus graphs. In this paper, we first characterize k-cactus graphs for 2≤k≤4 based on the block decompositions. Subsequently, we give tight upper bounds on their sizes. Moreover, the corresponding extremal graphs are also characterized. However, for larger k, this extremal problem remains unsolved. Besides, we prove that every 2-connected k-cactus (k≥1) with n vertices has at most n+k−1 edges, and the bound is tight if n≥k+2.