2017/05/02 by Saeid Alikhani, Alikhani, Saeid, Nasrin Jafari +1
Computer Science · Mathematics · #05C30 #05C69 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Graph theory and applications
paper · pdf · doi:10.48550/arxiv.1705.00826
openalex publication_date 2017/05/02 · openalex created_date 2017/05/12 · openalex updated_date 2026/07/28
Let G = (V, E) be a simple graph of order n. The total dominating set of G is a subset D of V that every vertex of V is adjacent to some vertices of D. The total domination number of G is equal to minimum cardinality of total dominating set in G and is denoted by γt(G). The total domination polynomial of G is the polynomial Dt(G,x)=∑i=γt(G)n dt(G,i)xi, where dt(G,i) is the number of total dominating sets of G of size i. A root of Dt(G,x) is called a total domination root of G. An irrelevant edge of Dt(G,x) is an edge e ∈ E, such that Dt(G, x) = Dt(G∖ e, x). In this paper, we characterize edges possessing this property. Also we obtain some results for the number of total dominating sets of a regular graph. Finally, we study graphs with exactly two total domination roots \-3,0\, \-2,0\ and \-1,0\.