Families of pairs of graphs with a large number of common cards

Abstract

Abstract:The vertex-deleted subgraph G−v, obtained from the graph G by deleting the vertex v and all edges incident to v, is called a card of G. The deck of G is the multiset of its unlabelled vertex-deleted subgraphs. The number of common cards of G and H (or between G and H) is the cardinality of the multiset intersection of the decks of G and H. In this article, we present infinite families of pairs of graphs of order n ≥ 4 that have at least 2 1 3 (n−1) common cards; we conjecture that these, along with a small number of other families constructed from them, are the only pairs of graphs having this many common cards, for sufficiently large n. This leads us to propose a new stronger version of the Reconstruction Conjecture. In addition, we present an infinite family of pairs of graphs with the same degree sequence that have (n−5) common cards, respectively. We then present new families that have the maximum number of common cards when one graph is connected and the other disconnected. Finally, we present a family with a large number of common cards, where one graph is a tree and the other unicyclic, and discuss how many cards are required to determine whether a graph is a tree.

Recognizing connectedness from vertex-deleted subgraphs

“…Then, in Section 3, we show that, with the exception of six pairs of graphs of order at most 7, any pair of graphs that attains the maximum is in one of four infinite families. This result establishes the truth of Conjecture 3.5 in [5].…”

“…Since the sufficiency of the condition is proved in Theorem 3.4 of [5], we only need prove that it is necessary.…”

The adversary degree-associated reconstruction number of double-brooms

“…Let y be the other neighbor of x in H 2 . If y is the one 2-vertex in B m, 3 , then G has only one 2-vertex. If y is the (m + 1)-vertex in H 2 , then deleting the only 2-vertex other than x leaves an isolated vertex, which does not occur in a middle dacard of D m,n,5 .…”

“…Bowler, Brown, and Fenner [3] constructed infinite families of pairs of graphs in which the pairs with n vertices have 2 (n − 4)/3 common dacards, so adrn(G) can be as large as 2 (n − 4)/3 + 1. They conjecture that this is the largest value for n-vertex graphs.…”

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 !…”