The degree-associated reconstruction number of an unicentroidal tree

Document Type : Research Paper

Authors

Department of Mathematics, Shahid Beheshti University, Tehran, Iran

10.22108/toc.2024.131563.1938

Abstract

As we know, by deleting one vertex of a graph $G$, we have a subgraph of $G$ called a card of $G$. Also, investigation of that each graph with at least three vertices is determined by its multiset of cards, is called the reconstruction conjecture and the minimum number of dacards that determine $G$ is denoted the degree-associated reconstruction number $drn(G)$. Barrus and West conjectured that $drn(G) \leq 2$ for all but finitely many trees. A tree is unicentroidal or bicentroidal when it has one or two centroids, respectively. An unicentroidal tree $T$ with centroid $v$ is symmetrical if for two neighbours of $u$ and $u'$ of $v$, there exists an automorphism on $T$ mapping $u$ to $u'$. In \cite{Shad}, Shadravan and Borzooei proved that the conjecture is true for any non-symmetrical unicentroidal tree. In this paper, we proved that for any symmetrical unicentroidal tree $T$, $drn(T) \leq 2$. So, we concluded that the conjecture is true for any unicentroidal tree.

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Articles in Press, Corrected Proof
Available Online from 10 February 2024
  • Receive Date: 20 November 2021
  • Revise Date: 21 September 2023
  • Accept Date: 22 January 2024
  • Published Online: 10 February 2024