Let $G=(V,E)$ be a graph. A subset $D\subseteq V$ is a dominating set if every vertex not in $D$ is adjacent to a vertex in $D$. A dominating set $D$ is called a total dominating set if every vertex in $D$ is adjacent to a vertex in $D$. The domination (resp. total domination) number of $G$ is the smallest cardinality of a dominating (resp. total dominating) set of $G$. The bondage (resp. total bondage) number of a nonempty graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with larger domination (resp. total domination) number of $G$. The reinforcement number of $G$ is the smallest number of edges whose addition to $G$ results in a graph with smaller domination number. This paper shows that the decision problems for bondage, total bondage and reinforcement are all NP-hard.
In this paper, we follow Xu [17] for graph-theoretical terminology and notation. A graph G = (V, E) always means a finite, undirected and simple graph, where V = V (G) is the vertex-set and E = E(G) is the edge-set of G.
A subset D ⊆ V is a dominating set of G if every vertex not in D is adjacent to a vertex in D. The domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set of G. A dominating set D is called a γ-set of G if |D| = γ(G). The bondage number of G, denoted by b(G), is the minimum number of edges whose removal from G results in a graph with larger domination number of G.
The reinforcement number of G, denoted by r(G), is the smallest number of edges whose addition to G results in a graph with smaller domination number of G. Domination is a classical concept in graph theory. The bondage number and the reinforcement number were introduced by Fink et at. [3] and Kok, Mynhardt [12], respectively, in 1990. The reinforcement number for digraphs has been studies by Huang, Wang and Xu [11]. Domination as well as related topics is now well studied in graph theory.
The literature on these subjects have been surveyed and detailed in the two excellent domination books by Haynes, Hedetniemi, and Slater [7,8].
Theory of domination has been applied in many research fields. For different applications, many variations of dominations were proposed in the research literature by adding some restricted conditions to dominating sets, for example, the total domination and the restrained domination.
A dominating set D is called a total dominating set if every vertex in D is adjacent to another vertex in D. The total domination number, denoted by γ t (G), of G is the minimum cardinality of a total dominating set of G. Use the symbol D t to denote a total dominating set. A total dominating set
The total bondage number of G, denoted by b t (G), is the minimum number of edges whose removal from G results in a graph with larger total domination number of G. The total domination was introduced by Cockayne et al. [1]. Total domination in graphs has been extensively studied in the literature. A survey of selected recent results on total domination in Henning [9]. The total bondage number of a graph was first studied by Kulli and Patwari [13] and further studied by Sridharan, Elias, Subramanian [15], Huang and Xu [10].
Analogously, a dominating set D is called a restrained dominating set if every vertex not in D is adjacent to another vertex not in D. The restrained domination number, denoted by γ r (G), of G is the minimum cardinality of a total dominating set of G. The restrained bondage number of G, denoted by b r (G), is the minimum number of edges whose removal from G results in a graph with larger restrained domination number of G. The restrained domination was introduced by Telle and Proskurowski [16], and the restrained bondage number was defined by Hattingh and Plummer [6].
Whys that a graph-theoretical parameter is proposed at once is to determine the exact value of this parameter for all graphs. However, the problem determining domination for general graphs has been proved to be NP-complete (see GT2 in Appendix in Garey and Johnson [4]); the problems determining total domination and restrained domination for general graphs have been also proved to be NP-complete by Laskar et al. [14], and by Domke et at. [2], respectively.
As regards the bondage problem, Hattingh et al. [6] showed that the restrained bondage problem is NP-complete even for bipartite graphs. For the general bondage problem, from the algorithmic point of view, Hartnell et at. [5] designed a linear time algorithm to compute the bondage number of a tree. However, the complexity of this problem is still unknown for other classes of graphs.
In this paper, we will show that the decision problems for bondage, total bondage and reinforcement are all NP-hard. Their proofs are Section 3, Section 4 and Section 2 3-satisfiability problem Following Garey and Johnson’s techniques for proving NP-hardness [4], we prove our results by describing a polynomial transformation from the known NP-complete problem: 3-satisfiability problem. To state the 3-satisfiability problem, we, in this section, recall some terms we will use in describing it.
Let U be a set of Boolean variables. A truth assignment for U is a mapping t :
then u is said to be" false" under t. If u is a variable in U, then u and ū are literals over U. The literal u is true under t if and only if the variable u is true under t; the literal ū is true if and only if the variable u is false.
A clause over U is a set of literals over U. It represents the disjunction of these literals and is satisfied by a truth assignment if and only if at least one of its members is true under that assignment. A collection C of clauses over U is satisfiable if and only if there exists some truth assignment for U that simultaneously satisfies all the clauses in C . Such a truth assignme
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