JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (6): 18-24, 39.doi: 10.6040/j.issn.1671-9352.0.2021.420

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The k-path vertex cover in some products graphs of star graph and bipartite graph

Huiling YIN(),Jingrong CHEN*(),Xiaoyan SU   

  1. School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China
  • Received:2021-06-09 Online:2023-06-20 Published:2023-05-23
  • Contact: Jingrong CHEN E-mail:1553408365@qq.com;chenjr@mail.lzjtu.cn

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Abstract:

For a subset $S \subset V(G)$, if any k-path contains at least one vertex from S, then it is called a k-path vertex cover set of the graph G. The minimum cardinality of k-path vertex cover set is called the k-path vertex cover number, which is denoted by ψk(G). The k-path vertex cover problem is studied for Cartesian product graphs, lexicographic product graphs, directed product graphs of star graph and bipartite graph, and obtain an upper and a lower bound by enumeration and related concepts of subgraphs.

Key words: k-path vertex cover, star graph, bipartite graph

CLC Number: 

  • O157.5
1 SELKOWS M .The independence number of graphs in terms of degrees[J].Discrete Mathematics,1993,122,343-348.
doi: 10.1016/0012-365X(93)90307-F
2 HARANTJ , SCHIERMEYERI .On the independence number of a graph in terms of order and size[J].Discrete Mathematics,2001,232,131-138.
doi: 10.1016/S0012-365X(00)00298-3
3 LIUXianliang , LUHongliang , WANGWei ,et al.PTAS for the minimum k-path connected vertex cover problem in unit disk graphs[J].Journal of Global Optimization,2013,56(2):449-458.
doi: 10.1007/s10898-011-9831-x
4 KARDOSF , KATRENICJ , SCHIERMEYERI .On computing the minimum 3-path vertex cover and dissociation number of graphs[J].Theoretical Computer Science,2011,412(50):7009-7017.
doi: 10.1016/j.tcs.2011.09.009
5 JAKOVACM , TARANENKOA .On the vertex k-path cover[J].Discrete Applied Mathematics,2013,161(13/14):19431949.
6 BREŠARB , KARDOŠF , KATRENIČJ ,et al.Minimum k-path vertex cover[J].Discrete Applied Mathematics,2011,159(12):1189-1195.
doi: 10.1016/j.dam.2011.04.008
7 JAKOVACM , TARANENKOA .On the k-path vertex cover of some graph products[J].Discrete Mathematics,2013,313(1):94-100.
doi: 10.1016/j.disc.2012.09.010
8 JAKOVACM .The $k$-path vertex cover of rooted product graphs[J].Discrete Applied Mathematics,2015,187(C):111-119.
9 张㢶㢷. 一些乘积图的k-路顶点覆盖[D]. 天津: 天津师范大学, 2016.
ZHANG Bitao. The k-path vertex cover of some product graphs[D]. Tianjin: Tianjin Normal University Press, 2016.
10 ZHANGBitao , ZUOLiancui .The k-path vertex cover of some product graphs[J].WSEAS Transactions on Mathematics,2016,15,374-384.
11 LIZhao , ZUOLiancui .The k-path vertex cover in Cartesian product graphs and complete bipartite graphs[J].Applied Mathematics and Computation,2018,331,69-79.
doi: 10.1016/j.amc.2018.03.008
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