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Characterization of graphs using domination polynomials. (English) Zbl 1207.05092
Summary: Let \(G\) be a simple graph of order \(n\). The domination polynomial of \(G\) is the polynomial \[ D(G,x)=\sum^n_{i=1} d(G,i)^i, \] where \(d(G,i)\) is the number of dominating sets of \(G\) of size \(i\). A root of \(D(G,x)\) is called a domination root of \(G\). We denote the set of distinct domination roots by \(Z(D(G,x))\). Two graphs \(G\) and \(H\) are said to be \({\mathcal D}\)-equivalent, written as \(G\sim H\), if \(D(G,x)= D(H,x)\). The \({\mathcal D}\)-equivalence class of \(G\) is \([G]= \{H: H\sim G\}\). A graph \(G\) is said to be \({\mathcal D}\)-unique if \([G]= \{G\}\).
In this paper, we show that if a graph \(G\) has two distinct domination roots, then \(Z(D(G,x))= \{-2, 0\}\). Also, if \(G\) is a graph with no pendant vertex and has three distinct domination roots, then \[ Z(D(G,x))\subseteq \Biggl\{0,-2\pm\sqrt{2}i,{-3+ \sqrt{3}i\over 2}\Biggr\}. \] Also, we study the \({\mathcal D}\)-equivalence classes of some certain graphs. It is shown that if \(n\equiv 0,2\pmod 3\), then \(C_n\) is \({\mathcal D}\)-unique, and if \(n\equiv 0\pmod 3\), then \([P_n]\) consists of exactly two graphs.

MSC:
05C31 Graph polynomials
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
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