Graphs with arbitrarily large adversary degree associated reconstruction number
P. Anusha Devi, S. Monikandan · AKCE International Journal of Graphs and Combinatorics · 2020
A vertex deleted unlabeled subgraph of a graph is a card. A dacard specifies the degree of the deleted vertex along with the card. The adversary degree associated reconstruction number of a graph G, denoted adrn(G), is the minimum number k such that every collection of k dacards of G uniquely determines G. A strong double broom is the graph on at least 5 vertices obtained from a union of (at least two) internally vertex disjoint paths with same ends u and v by appending leaves at u and v. The strong double broom, obtained from a union of m internally vertex disjoint paths of order k with same ends u and v by appending n leaves at each u and v, is denoted by B(n,n,mPk). In this paper, we show that the drn of all strong double brooms is two and we determine adrn(B(n,n,mPk)) for all n, m, k. For n≥1 and m≥2, usually adrn(B(n,n,mP3))=n+2 and adrn(B(n,n,mP4))=n+m+2, except adrn(B(1,1,2P3))=4. For n≥1,m≥2 and k>4, usually adrn(B(n,n,mPk))=n+2, except adrn(B(n,n,mP5))=max{m,n}+2 when (n,m)≠(1,2) and adrn(B(1,1,2P5))=5.