Abstract
The weighted independent set problem on P 5 -free graphs has numerous applications, including data mining and dispatching in railways. The recognition of P 5 -free graphs is executed in polynomial time. Many problems, such as chromatic number and dominating set, are NP-hard in the class of P 5 -free graphs. The size of a minimum independent feedback vertex set that belongs to a P 5 -free graph with n vertices can be computed in O ( n 16 ) time. The unweighted problems, clique and clique cover, are NP-complete and the independent set is polynomial. In this work, the P 5 -free graphs using the weak decomposition are characterized, as is the dominating clique, and they are given an O ( n ( n + m ) ) recognition algorithm. Additionally, we calculate directly the clique number and the chromatic number; determine in O ( n ) time, the size of a minimum independent feedback vertex set; and determine in O ( n + m ) time the number of stability, the dominating number and the minimum clique cover.
Highlights
From Consequence 2 it follow that the clique number and the chromatic number are calculated directly; the number of stability is determined in O(n + m); the minimum clique cover and the dominating number are O(n + m) (since the determination of the neighbors of a vertex in (B or A − B) is not more than the complexity of the weak decomposition algorithm)
In this paper the P5 -free graphs are characterized using the weak decomposition presented in Theorem 10
A result of Consequence 1 is the direct calculation of the clique and chromatic number of the P5 -free graphs
Summary
Graphs, including the P5 -free graphs, have many real-life applications, including: preference elicitation applied to a brownfield redevelopment conflict in China [1], evaluation of the energy supply options of a manufacturing plant [2], lifestyle pattern mining based on image collections in smartphones [3] and conflict resolution based on option prioritization [4]. With G − X we denote the graph G (V − X ), every time X ⊆ V, and we write G − v, (∀v ∈ V), when X = {v}. If v ∈ V is a vertex in G, the neighborhood NG (v) represents the vertices of G − v that are adjacent to v. N (v) denotes the neighborhood of the vertex v in the complement of the graph G. A clique represents a subset of V in that all the vertices are pairwise adjacent. The stability number α( G ) of a graph G is the size of the greater stable set. An independent (stable) set of a graph G is a subset of pairwise non-adjacent vertices. A graph denoted G is F-free in the case that none of its induced subgraphs are in F.
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