On the outer-connected reinforcement and bondage problems in bipartite graphs: the algorithmic complexity

Abstract

An outer connected dominating(OCD) set of a graph $G=(V,E)$ is a set $tilde{D} subseteq V$ such that every vertex not in $S$ is adjacent to a vertex in $S$, and the induced subgraph of $G$ by $V setminus tilde{D}$, i.e. $G [V setminus tilde{D}]$, is connected. The OCD number of $G$ is the smallest cardinality of an OCD set of $G$. The outer-connected bondage number of a nonempty graph G is the smallest number of edges whose removal from G results in a graph with a larger OCD number. Also, the outer-connected reinforcement number of G is the smallest number of edges whose addition to G results in a graph with a smaller OCD number. In 2018, Hashemi et al. demonstrated that the decision problems for the Outer-Connected Bondage and the Outer-Connected Reinforcement numbers are all NP-hard in general graphs. In this paper, we improve these results and show their hardness for bipartite graphs. Also, we obtain bounds for the outer-connected bondage number.