# The complexity of the Bondage problem in planar graphs

1 CEDRIC - OC - CEDRIC. Optimisation Combinatoire
CEDRIC - Centre d'études et de recherche en informatique et communications
2 CEDRIC - ROC - CEDRIC. Réseaux et Objets Connectés
CEDRIC - Centre d'études et de recherche en informatique et communications
Abstract : A set $S\subseteq V(G)$ of a graph $G$ is a dominating set if each vertex has a neighbor in $S$ or belongs to $S$. Let $\gamma(G)$ be the cardinality of a minimum dominating set in $G$. The bondage number $b(G)$ of a graph $G$ is the smallest number of edges $A\subseteq E(G)$, such that $\gamma(G-A)=\gamma(G)+1$. The problem of finding $b(G)$ for a graph $G$ is known to be NP-hard even for bipartite graphs. In this paper, we show that deciding if $b(G)=1$ is NP-hard, while deciding if $b(G)=2$ is coNP-hard, even when $G$ is restricted to one of the following classes: planar $3$-regular graphs, planar claw-free graphs with maximum degree $3$, planar bipartite graphs of maximum degree $3$ with girth $k$, for any fixed $k\geq 3$.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-03299625
Contributor : Valentin Bouquet Connect in order to contact the contributor
Submitted on : Monday, July 26, 2021 - 3:49:20 PM
Last modification on : Friday, August 5, 2022 - 2:54:01 PM

### Identifiers

• HAL Id : hal-03299625, version 1
• ARXIV : 2107.11216

### Citation

Valentin Bouquet. The complexity of the Bondage problem in planar graphs. 2021. ⟨hal-03299625⟩

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