Graphs that are simultaneously efficient open domination and efficient closed domination graphs

TL;DR

This paper investigates EOCD graphs, which have vertex subsets partitioning their neighborhoods, explores their structure, proves the NP-completeness of recognizing them, and characterizes them in specific graph classes.

Contribution

It provides a structural analysis of EOCD graphs, proves recognition is NP-complete, and offers recursive constructions and characterizations within certain graph families.

Findings

EOCD graphs are characterized among Sierpiński graphs.

Recognition of EOCD graphs is NP-complete.

A recursive method constructs all EOCD trees.

Abstract

A graph is an efficient open (resp.\ closed) domination graph if there exists a subset of vertices whose open (resp.\ closed) neighborhoods partition its vertex set. Graphs that are efficient open as well as efficient closed (shortly EOCD graphs) are investigated. The structure of EOCD graphs with respect to their efficient open and efficient closed dominating sets is explained. It is shown that the decision problem regarding whether a graph is an EOCD graph is an NP-complete problem. A recursive description that constructs all EOCD trees is given and EOCD graphs are characterized among the Sierpi\'nski graphs.