The k-path vertex cover: general bounds and chordal graphs

Abstract

For an integer k ≥ 3, a k-path vertex cover of a graph G = (V, E) is a set T ⊆ V that shares a vertex with every path subgraph of order k in G. The minimum cardinality of a k-path vertex cover is denoted by ψ k (G). We give estimates -mostly upper bounds -on ψ k (G) in terms of various parameters, including vertex degrees and the number of vertices and edges. The problem is also considered on chordal graphs and planar graphs.

A Survey on the k-Path Vertex Cover Problem

“…Recently, Bujtás et al [16] generalized the result to arbitrary graphs in terms of minimum and maximum degree.…”

“…Theorem 6.7. [16] Let k and ∆ be integers with k ≥ 3 and ∆ = 2 or ∆ ≥ 4, and let G be a graph of maximum degree at most ∆. Then the following hold.…”

“…Theorem 6.8. [16] Let G be a graph without isolated vertices. Then for every integer k ≥ 3, the following holds.…”

“…Theorem 6.10. [12,16] Let G be a graph with n vertices and m edges. Then (i) ψ 3 (G) ≤ max{ 2n+m 6 , n+m 4 , 4n+m 9 }, (ii) ψ 4 (G) ≤ n+3m 10 .…”

“…Theorem 6.14. [16] Let G be a planar graph of order n, then ψ 3 (G) ≤ 11n/15 and ψ 6 (G) ≤ 2n/3. Theorem 6.15.…”