Connected and outer-connected domination number of middle graphs

TL;DR

This paper investigates the connected and outer-connected domination numbers of middle graphs, providing bounds, explicit calculations for specific graph families, and Nordhaus-Gaddum-like relations.

Contribution

It introduces new bounds and exact values for the outer-connected domination number of middle graphs for various graph families and operations.

Findings

Derived tight bounds in terms of graph order

Computed outer-connected domination numbers for specific graph families

Established Nordhaus-Gaddum-like relations for middle graphs

Abstract

In this paper, we study the notions of connected domination number and of outer-connected domination number for middle graphs. Indeed, we obtain tight bounds for this number in terms of the order of the graph M(G). We also compute the outer-connected domination number of some families of graphs such as star graphs, cycle graphs, wheel graphs, complete graphs, complete bipartite graphs and some operation on graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the outer-connected domination number of middle graphs.