Affiliation:
1. Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education School of Mathematics and Computer Science Xiangtan University Xiangtan Hunan China
2. School of Mathematical Sciences Laboratory of Mathematics and Complex Systems, Ministry of Education Beijing Normal University Beijing China
Abstract
AbstractA generalized ‐independent set is a set of vertices such that the induced subgraph contains no trees with ‐vertices, and the generalized ‐independence number is the cardinality of a maximum ‐independent set in . Zito proved that the maximum number of maximum generalized 2‐independent sets in a tree of order is if is odd, and if is even. Tu et al. showed that the maximum number of maximum generalized 3‐independent sets in a tree of order is if , and if , and if and they characterized all the extremal graphs. Inspired by these two nice results, we establish four structure theorems about maximum generalized ‐independent sets in a tree for a general integer . As applications, we show that the maximum number of generalized 4‐independent sets in a tree of order is
and we also characterize the structure of all extremal trees with the maximum number of maximum generalized 4‐independent sets.
Funder
National Natural Science Foundation of China
Reference21 articles.
1. NP-hard graph problems and boundary classes of graphs
2. An improved algorithm for the vertex cover P3 ${P}_{3}$ problem on graphs of bounded treewidth;Bai Z.;Discrete Math. Theoret. Comput. Sci,2019
3. On computing the dissociation number and induced matching number of bipartite graphs;Bolic Z.;Ars Combin,2004
4. Minimum k-path vertex cover
5. Independent packings in structured graphs