Graphs with arbitrarily large adversary degree associated reconstruction number

Abstract

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 adrnðGÞ, 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 Bðn, n, mP k Þ: In this paper, we show that the drn of all strong double brooms is two and we determine adrnðBðn, n, mP k ÞÞ for all n, m, k: For n ! 1 and m ! 2, usually adrnðBðn, n, mP 3 ÞÞ ¼ n þ 2 and adrnðBðn, n, mP 4 ÞÞ ¼ n þ m þ 2, except adrnðBð1, 1, 2P 3 ÞÞ ¼ 4: For n ! 1, m ! 2 and k > 4, usually adrnðBðn, n, mP k ÞÞ ¼ n þ 2, except adrnðBðn, n, mP 5 ÞÞ ¼ maxfm, ng þ 2 when ðn, mÞ 6 ¼ ð1, 2Þ and adrnðBð1, 1, 2P 5 ÞÞ ¼ 5: KEYWORD