On Making a Distinguished Vertex of Minimum Degree by Vertex Deletion

Abstract

Abstract. For directed and undirected graphs, we study the problem to make a distinguished vertex the unique minimum-(in)degree vertex through deletion of a minimum number of vertices. The corresponding NP-hard optimization problems are motivated by applications concerning control in elections and social network analysis. Continuing previous work for the directed case, we show that the problem is W[2]-hard when parameterized by the graph's feedback arc set number, whereas it becomes fixed-parameter tractable when combining the parameters "feedback vertex set number" and "number of vertices to delete". For the so far unstudied undirected case, we show that the problem is NP-hard and W[1]-hard when parameterized by the "number of vertices to delete". On the positive side, we show fixed-parameter tractability for several parameterizations measuring tree-likeness, including a vertex-linear problem kernel with respect to the parameter "feedback edge set number". On the contrary, we show a non-existence result concerning polynomial-size problem kernels for the combined parameter "vertex cover number and number of vertices to delete", implying corresponding nonexistence results when replacing vertex cover number by treewidth or feedback vertex set number.

On Structural Parameterizations of the Bounded-Degree Vertex Deletion Problem

“…This paper studies the Bounded-degree Vertex deletion problem (BDD): given an undirected graph G, a degree bound d, and a limit , determine whether it is possible to delete at most vertices from G in order to obtain a graph of maximum degree at most d. Aside from being a natural generalization of the classical Vertex CoVer problem, BDD has found applications in areas such as computational biology [19] and is the dual problem of the so-called s-Plex Detection problem in social network analysis [3,38,39,44]. Finally, related problems on directed as well as undirected graphs which model problems in voting theory and social network analysis have also been studied in the literature [5,7].…”

On the complexity of making a distinguished vertex minimum or maximum degree by vertex deletion

“…Variants of these problems include the weighted case, in which we are interested in finding a vertex set S of minimum weight instead of minimum cardinality, when each vertex in G has a weight associated with it. These problems have been previously studied in [1,2] with reference to directed graphs and electoral networks. The most natural motivation lies in competitive social networks, which are undirected, and in which the degree of a node is widely seen as a measure of its popularity, influence or importance.…”

“…Both MDD(min) and MDD(max) are known to be NP-complete [1]. Previous work on these two problems involved approaches using parameterized complexity [1], but a classical complexity approach has not yet been taken as per our knowledge.…”

“…Both MDD(min) and MDD(max) are known to be NP-complete [1]. Previous work on these two problems involved approaches using parameterized complexity [1], but a classical complexity approach has not yet been taken as per our knowledge. In this paper, we take a classical complexity theory approach towards the problems and make the following contributions:…”

“…Given a vertex weighted graph G = (V, E) and a specified, or "distinguished" vertex p ∈ V , MDD(min) is the problem of finding a minimum weight vertex set S ⊆ V \ {p} such that p becomes the minimum degree vertex in G[V \ S]; and MDD(max) is the problem of finding a minimum weight vertex set S ⊆ V \{p} such that p becomes the maximum degree vertex in G[V \ S]. These are known NPcomplete problems and have been studied from the parameterized complexity point of view in [1]. Here, we prove that for any ǫ > 0, both the problems cannot be approximated within a factor (1 − ǫ) log n, unless NP ⊆ Dtime(n log log n ).…”