W[1]-hardness of Outer Connected Dominating set in d-degenerate Graphs

TL;DR

This paper investigates the computational complexity of finding minimum outer-connected dominating sets in sparse graphs, proving it is W[1]-hard on d-degenerate graphs, contrasting with the fixed-parameter tractability of related problems.

Contribution

It establishes the W[1]-hardness of the outer-connected domination problem on d-degenerate graphs, highlighting its computational difficulty in sparse graph classes.

Findings

Outer-connected domination is W[1]-hard on d-degenerate graphs.

Connected dominating set has an FPT algorithm on d-degenerate graphs.

NP-complete even for bipartite graphs.

Abstract

A set $D \subseteq V$ of a graph $G = (V,E)$ is called an outer-connected dominating set of $G$ if every vertex $v$ not in $D$ is adjacent to at least one vertex in $D$, and the induced subgraph of $G$ on $V \setminus D$ is connected. The Minimum Outer-connected Domination problem is to find an outer-connected dominating set of minimum cardinality for the input graph $G$. Given a positive integer $k$ and a graph $G = (V, E)$, the Outer-connected Domination Decision problem is to decide whether $G$ has an outer-connected dominating set of cardinality at most $k$. The Outer-connected Domination Decision problem is known to be NP-complete, even for bipartite graphs. We study the problem of outer-connected domination on sparse graphs from the perspective of parameterized complexity and show that it is W[1]-hard on d-degenerate graphs, while the original connected dominating set has FTP algorithm on d-degenerate graphs.