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Communication Dans Un Congrès Année : 2023

Identifying codes in bipartite graphs of given maximum degree

Résumé

An identifying code of a closed-twin-free graph G is a set S of vertices of G such that any two vertices in G have a distinct intersection between their closed neighborhoods and S. It was conjectured in [F. Foucaud, R. Klasing, A. Kosowski, A. Raspaud. On the size of identifying codes in triangle-free graphs. Discrete Applied Mathematics, 2012] that there exists an absolute constant c such that for every connected graph G of order n and maximum degree ∆, G admits an identifying code of size at most ∆−1 ∆ n + c. We provide significant support for this conjecture by proving it for the class of all bipartite graphs that do not contain any pairs of open-twins of degree at least 2. In particular, this class of bipartite graphs contains all trees and more generally, all bipartite graphs without 4-cycles. Moreover, our proof allows us to precisely determine the constant c for the considered class, and the list of graphs needing c > 0. For ∆ = 2 (the graph is a path or a cycle), it is long known that c = 3/2 suffices. For connected graphs in the considered graph class, for each ∆ ≥ 3, we show that c = 1/∆ ≤ 1/3 suffices and that c is required to be positive only for a finite number of trees. In particular, for ∆ = 3, there are 12 trees with diameter at most 6 with a positive constant c and, for each ∆ ≥ 4, the only tree with positive constant c is the ∆-star. Our proof is based on induction and utilizes recent results from [F. Foucaud, T. Lehtilä. Revisiting and improving upper bounds for identifying codes. SIAM Journal on Discrete Mathematics, 2022].
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Dates et versions

hal-04213143 , version 1 (21-09-2023)

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Dipayan Chakraborty, Florent Foucaud, Tuomo Lehtilä. Identifying codes in bipartite graphs of given maximum degree. 12th Latin-American Algorithms, Graphs and Optimization Symposium (LAGOS 2023), Sep 2023, Huatulco, Oaxaca, México, Mexico. pp.157-165, ⟨10.1016/j.procs.2023.08.225⟩. ⟨hal-04213143⟩
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