On the Distance Paired-Domination of Circulant Graphs

Let G=(V,E) be a graph without isolated vertices. A set D⊆V is a d-distance paired-dominating set of G if D is a d-distance dominating set of G and the induced subgraph G=(V,E) has a perfect matching. The minimum cardinality of a d-distance paired-dominating set for graph G is the d-distance paired-domination number, denoted by γp^d(G). In this paper, we study the d-distance paired-domination number of circulant graphs C(n;{1,k}) for 2≤k≤4. We prove that for k=2, n≥5 and d≥1,γp^d(C(n;{1,k}))=2[n/2kd+3],for k=3, n≥7 and d≥1,γp^d(C(n;{1,k}))=2[n/2kd+2],and for k=4 and n≥9,(i) if d=1, thenγp(C(n;{1,k}))={2[3n/23]+2,2[3n/23],if n≡15,22 (mod 23);otherwise (ii) if d≥2, thenγp^d(C(n;{1,k}))={2[2n/4kd+1]+2,2[2n/4kd+1],if n≡2kd,4kd−1,4kd (mod 4kd+1)otherwise.