The adversary degree-associated reconstruction number of double-brooms

Abstract

A vertex-deleted subgraph of a graph G is a card. A dacard specifies the degree of the deleted vertex along with the card. The adversary degree-associated reconstruction number adrn(G) is the least k such that every set of k dacards determines G. We determine adrn(D m,n,p ), where the double-broom D m,n,p with p ≥ 2 is the tree with m + n + p vertices obtained from a path with p vertices by appending m leaves at one end and n leaves at the other end. We determine adrn(D m,n,p ) for all m, n, p. For 2 ≤ m ≤ n, usually adrn(D m,n,p ) = m + 2, except adrn(D m,m+1,p ) = m + 1 and adrn (D m,m+2,p There are exceptions when (m, n) = (2, 3) or p = 4. For m = 1 the usual value is 4, with exceptions when p ∈ {2, 3} or n = 2.

Degree associated edge reconstruction number of split graphs with biregular independent set is one

“…Ramachandran [19,20] then studied the degree associated reconstruction number of graphs and digraphs in 2000. The degree (degree triple) associated reconstruction number of a graph (digraph) D is the size of the smallest collection of dacards of D that uniquely determines D. Articles [1,3] and [11] are recent papers on this parameter.…”

Degree Associated Edge Reconstruction Parameters of Strong Double Brooms

“…Ramachandran [19,20] then studied the degree associated reconstruction number of graphs and digraphs in 2000. The degree (degree triple) associated reconstruction number of a graph (digraph) D is the size of the smallest collection of dacards of D that uniquely determines D. Articles [1], [2], [3], [6] and [13] are recent papers on the degree associated reconstruction number.…”

“…Conjecture that remains open. For very few classes of graphs, these edge reconstruction parameters have been determined [4,12,13,16,17].…”

“…Recently Ma et al [13] have determined adrn of double brooms. In this paper, we determine dern and adern of strong double brooms.…”

Graphs with arbitrarily large adversary degree associated reconstruction number

“…) Myrvold [11] showed arn(G) ¼ 3 for almost every graph G. Since always adrnðGÞ arnðGÞ, it is thus of some interest to find graphs G where adrn(G) is large. Bowler et al [2] constructed infinite families of pairs of graphs in which the pairs with n vertices have 2b nÀ4 3 c common dacards, so adrn(G) can be as large as 2b nÀ4 3 c þ 1: They conjecture that this is the largest value for graphs of order n. Among vertex-transitive graphs, G ¼ 2K n 4 , n 4 in [1] achieves adrnðGÞ ¼ drnðGÞ ¼ 1 4 jVðGÞj þ 2: Recently Ma et al [7] have proved that the adrn of most of the double brooms Bðm, n, P k Þ is minfm, ng þ 2: 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 !…”