On the sizes of generalized cactus graphs

TL;DR

This paper explores the maximum sizes of generalized cactus graphs, establishing bounds for specific cases and characterizing extremal graphs, while highlighting open problems for larger values of k.

Contribution

It characterizes k-cactus graphs for 2≤k≤4, provides tight upper bounds on their sizes, and extends results to 2-connected graphs for all positive integers k.

Findings

Maximum edges in 2-connected k-cactus graphs is n+k-1 for n≥k+2.

Characterization of extremal graphs for k-cactus with 2≤k≤4.

Open problems for k≥5 and bounds for n<k+1.

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 $k\ge 1$. It is well known that every cactus with $n$ vertices has at most $\lfloor\frac{3}{2}(n-1) \rfloor$ 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 $2\le k\le 4$ based on the block decompositions. Subsequently, we give tight upper bounds on their sizes. Moreover, the corresponding extremal graphs are also characterized. However, the case of $k\ge 5$ remains open. For the case of 2-connectedness, the range of $k$ is expanded to all positive integers in our research. We prove that every $2$-connected $k ~(\ge 1)$-cactus graphs with $n$ vertices has at most $n+k-1$ edges, and the bound is tight if $n \ge k + 2$. But, for $n < k+1$, determining best bounds remains a mystery except for some small values of $k$.