On maximizing private neighbors in graphs

Hedetniemi, Stephen T.; Rall, Douglas

doi:10.22049/cco.2026.31182.2771

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	On maximizing private neighbors in graphs
Communications in Combinatorics and Optimization
مقاله 17، دوره 11، شماره 4، اسفند 2026، صفحه 1265-1279 اصل مقاله (454.52 K)
نوع مقاله: Original paper
شناسه دیجیتال (DOI): 10.22049/cco.2026.31182.2771
نویسندگان
Stephen T. Hedetniemi¹؛ Douglas Rall^* ²
¹Emeritus Professor of Computer Science, Clemson University, Clemson, SC, USA
²Emeritus Professor of Mathematics, Furman University, Greenville, SC, USA
چکیده
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.
کلیدواژه‌ها
private neighbor؛ irredundance؛ domination

مراجع
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On maximizing private neighbors in graphs