Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 11 4 2026 12 01 On maximizing private neighbors in graphs 1265 1279 15181 10.22049/cco.2026.31182.2771 EN Stephen T. Hedetniemi Emeritus Professor of Computer Science, Clemson University, Clemson, SC, USA Douglas Rall Emeritus Professor of Mathematics, Furman University, Greenville, SC, USA Journal Article 2025 11 03 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. https://comb-opt.azaruniv.ac.ir/article_15181_ecb50e26f7f3be72f147f9eb97b99bc4.pdf