Total isolation of k-cliques in a graph

• Published in: Discrete mathematics Vol. 348; no. 12; p. 114689
• Main Authors: Cao, Yupei, An, Xinhui, Wu, Baoyindureng
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.12.2025
• Subjects: Clique; Total domination number; Total isolating set; Total isolating set; Clique; Total domination number
• ISSN: 0012-365X
• DOI: 10.1016/j.disc.2025.114689

For a graph G=(V(G),E(G)) and any positive integer k, a set D⊆V(G) is called a total k-clique isolating set of G if G−N[D] contains no k-clique and D induces a subgraph with no vertex of degree 0. The total k-clique isolation number ιt(G,k) is the minimum cardinality of a total k-clique isolating set of G. Clearly, ιt(G,1) is the total domination number of G, and ιt(G,2) was investigated by Boyer, Goddard and Henning recently. In this paper, we prove that for k≥3 and n≥k+2, if G is a connected graph of order n, then ιt(G,k)≤2nk+2. The bound is sharp.