2015/03/17 by Xuezheng Lv, Lv, Xuezheng, Baoyindureng Wu +1
Computer Science · Mathematics · Neuroscience · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Nuclear Receptors and Signaling #math.CO
paper · pdf · doi:10.48550/arxiv.1503.04939
17 pages
arxiv created 2015/03/17 · openalex publication_date 2015/03/17 · arxiv updated 2015/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A subset S⊆ V in a graph G=(V,E) is a total [1,2]-set if, for every vertex v∈ V, 1≤ |N(v)∩ S|≤ 2. The minimum cardinality of a total [1,2]-set of G is called the total [1,2]-domination number, denoted by γt[1,2](G). We establish two sharp upper bounds on the total [1,2]-domination number of a graph G in terms of its order and minimum degree, and characterize the corresponding extremal graphs achieving these bounds. Moreover, we give some sufficient conditions for a graph without total [1,2]-set and for a graph with the same total [1,2]-domination number, [1,2]-domination number and domination number.