Nordhaus-Gaddum inequalities for the number of connected induced subgraphs in graphs

TL;DR

This paper investigates Nordhaus-Gaddum type inequalities for the number of connected induced subgraphs, characterizing extremal graphs that minimize or maximize the sum of these counts in a graph and its complement.

Contribution

It provides exact characterizations of graphs minimizing or maximizing the sum of connected induced subgraphs and their complements, including trees, unicyclic graphs, and graphs with specific properties.

Findings

Minimum sum achieved by graphs with no induced path on four vertices.

Maximum sum achieved by specific trees with degree sequence (ceil(n/2), floor(n/2), 1,...,1).

Graphs with maximum sum have diameter at most 3, no cut vertex, and connected complement.

Abstract

Let $\eta(G)$ be the number of connected induced subgraphs in a graph $G$, and $\overline{G}$ the complement of $G$. We prove that $\eta(G)+\eta(\overline{G})$ is minimum, among all $n$-vertex graphs, if and only if $G$ has no induced path on four vertices. Since the $n$-vertex star $S_n$ with maximum degree $n-1$ is the unique tree of diameter $2$, $\eta(S_n)+\eta(\overline{S_n})$ is minimum among all $n$-vertex trees, while the maximum is shown to be achieved only by the tree whose degree sequence is $(\lceil n/2\rceil,\lfloor n/2\rfloor,1,\dots,1)$. Furthermore, we prove that every graph $G$ of order $n\geq 5$ and with maximum $\eta(G)+\eta(\overline{G})$ must have diameter at most $3$, no cut vertex and the property that $\overline{G}$ is also connected. In both cases of trees and graphs that have the same order, we find that if $\eta(G)$ is maximum then $\eta(G)+\eta(\overline{G})$ is minimum. As corollaries to our results, we characterise the unique connected graph $G$ of given order and number of vertices of degree $1$, and the unique unicyclic (connected and has only one cycle) graphs $G$ of a given order that minimises $\eta(G)+\eta(\overline{G})$.