Using edge contractions to reduce the semitotal domination number

TL;DR

This paper investigates how contracting a limited number of edges can reduce the semitotal domination number of a graph, providing bounds, characterizations, and complexity results for the problem.

Contribution

It establishes that at most three edge contractions suffice to reduce the semitotal domination number and characterizes graphs based on the number of contractions needed, along with complexity classifications.

Findings

At most 3 edge contractions are needed to reduce the semitotal domination number.

Characterizations of graphs requiring 1, 2, or 3 contractions.

Complexity dichotomy for the problem in monogenic classes.

Abstract

In this paper, we consider the problem of reducing the semitotal domination number of a given graph by contracting $k$ edges, for some fixed $k \geq 1$. We show that this can always be done with at most 3 edge contractions and further characterise those graphs requiring 1, 2 or 3 edge contractions, respectively, to decrease their semitotal domination number. We then study the complexity of the problem for $k=1$ and obtain in particular a complete complexity dichotomy for monogenic classes.