Partial vertex covers and the complexity of some problems concerning static and dynamic monopolies

by   Hossein Soltani, et al.

Let G be a graph and τ be an assignment of nonnegative integer thresholds to the vertices of G. Denote the average of thresholds in τ by τ̅. A subset of vertices D is said to be a τ-dynamic monopoly, if V(G) can be partitioned into subsets D_0, D_1, ..., D_k such that D_0=D and for any i∈{0, ..., k-1}, each vertex v in D_i+1 has at least τ(v) neighbors in D_0∪...∪ D_i. Denote the size of smallest τ-dynamic monopoly by dyn_τ(G). Also a subset of vertices M is said to be a τ-static monopoly (or simply τ-monopoly) if any vertex v∈ V(G)∖ M has at least τ(v) neighbors in M. Denote the size of smallest τ-monopoly by mon_τ(G). For a given positive number t, denote by Sdyn_t(G) (resp. Smon_t(G)), the minimum dyn_τ(G) (resp. mon_τ(G)) among all threshold assignments τ with τ≥ t. In this paper we consider the concept of partial vertex cover as follows. Let G=(V, E) be a graph and t be any positive integer. A subset S⊆ V is said to be a t-partial vertex cover of G, if S covers at least t edges of G. Denote the smallest size of a t-partial vertex cover of G by Pβ_t(G). Let ρ, 0<ρ<1 be any fixed number and G be a given bipartite graph with m edges. We first prove that to determine the smallest cardinality of a set S⊆ V(G) such that S covers at least ρ m edges of G, is an NP-hard problem. Then we prove that for any constant t, Sdyn_t(G)=Pβ_nt-m(G) and Smon_t(G)=Pβ_nt/2(G), where n and m are the order and size of G, respectively.


page 1

page 2

page 3

page 4


On the fixed-parameter tractability of the partial vertex cover problem with a matching constraint in edge-weighted bipartite graphs

In the classical partial vertex cover problem, we are given a graph G an...

On Reconfigurability of Target Sets

We study the problem of deciding reconfigurability of target sets of a g...

Weak dynamic monopolies in social graphs

Dynamic monopolies were already defined and studied for the formulation ...

Kernelization for Partial Vertex Cover via (Additive) Expansion Lemma

Given a graph and two integers k and ℓ, Partial Vertex Cover asks for a ...

Dynamic monopolies for interval graphs with bounded thresholds

For a graph G and an integer-valued threshold function τ on its vertex s...

Minimum T-Joins and Signed-Circuit Covering

Let G be a graph and T be a vertex subset of G with even cardinality. A ...

Angle Covers: Algorithms and Complexity

Consider a graph with a rotation system, namely, for every vertex, a cir...

Please sign up or login with your details

Forgot password? Click here to reset