On maximizing private neighbors in graphs

TL;DR

This work introduces a comprehensive framework of private-neighbor invariants in graphs, defining seven maximization parameters (including $SPN(G)$, $IPN(G)$, $EPN(G)$, $ESPN(G)$, $EISPN(G)$, $EIPN(G)$, $ISPN(G)$) based on three private-neighbor types (self, internal, external) and explores their connections to classical notions such as independence, strong matchings, irredundance, and domination. It provides exact, closed-form values for these invariants on several graph families (paths, cycles, complete bipartite graphs, and grids), and examines efficient- and private-dominating structures through generalized efficient graphs and related constructions. The paper also establishes a sharp lower bound for $IPN(T)$ on trees, analyzes perfect and total perfect domination variants, and introduces upper private domination; it concludes with open problems and conjectures, including computational questions for IPN-type invariants and grid/tree behavior. Overall, the work broadens the toolkit for analyzing domination- and irredundance-related properties in graphs and offers concrete formulas, constructions, and directions for future research and applications in network design and combinatorial optimization.

Abstract

Given a set $U \subset V$ of vertices in a graph $G = (V, E)$, a {\it private neighbor with respect to the set $U$} is any vertex $w \in V$ having precisely one neighbor, say $v$, in $U$. If $w \in V - U$, then $w$ is called an {\it external private neighbor} of $v$ with respect to $U$. If $w \in U$ then $w$ is called an {\it internal private neighbor} of $v$ with respect to $U$. We also add one special case: if $w \in U$ and $N(w) \cap U = \emptyset$, then we say that $w$ is a {\it self private neighbor} with respect to $U$. By definition, a self private neighbor with respect to $U$ is an isolated vertex in the subgraph of $G$ induced by $U$. In this paper we consider the general problems of trying to find sets of vertices which maximize the number of private neighbors of specific types in a graph. In the process of doing this we define several new maximization parameters of graphs which generalize some known and well-studied parameters of graphs relating to vertex and edge independence, domination and irredundance in graphs.

Figures (3)

• Figure 1: Maximizing private neighbors on $P_8 {\,\Box\,} P_3$
• Figure 2: $IPN(G)=6$ and $2\alpha^*(G)=4$
• Figure 3: The tree $T_4$.

Theorems & Definitions (25)

• Proposition 1
• Proposition 2
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof
• Proposition 6
• proof