Cut and pendant vertices and the number of connected induced subgraphs of a graph

TL;DR

This paper investigates the maximum number of connected induced subgraphs in graphs with specified cut vertices and characterizes extremal structures, revealing that cycles minimize such subgraphs among cut vertex-free graphs.

Contribution

It provides exact maximum counts and characterizations for graphs with given order and cut vertices, and extends results to graphs with pendant vertices, highlighting extremal structures.

Findings

Cycle graphs have the fewest connected induced subgraphs among cut vertex-free graphs.

The maximum number of connected induced subgraphs is determined for graphs with given order and cut vertices.

Extremal graphs are characterized for given order and pendant vertices, with trees being minimal structures.

Abstract

A vertex whose removal in a graph $G$ increases the number of components of $G$ is called a cut vertex. For all $n,c$, we determine the maximum number of connected induced subgraphs in a connected graph with order $n$ and $c$ cut vertices, and also characterise those graphs attaining the bound. Moreover, we show that the cycle has the smallest number of connected induced subgraphs among all cut vertex-free connected graphs. The general case $c>0$ remains an open task. We also characterise the extremal graph structures given both order and number of pendant vertices, and establish the corresponding formulas for the number of connected induced subgraphs. The `minimal' graph in this case is a tree, thus coincides with the structure that was given by Li and Wang~[Further analysis on the total number of subtrees of trees. \emph{Electron. J. Comb.} 19(4), #P48, 2012].