Identifying codes in graphs of given maximum degree: Characterizing trees
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2024
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| author | Chakraborty, Dipayan Foucaud, Florent Henning, Michael A. Lehtilä, Tuomo |
| author_facet | Chakraborty, Dipayan Foucaud, Florent Henning, Michael A. Lehtilä, Tuomo |
| contents | An identifying code of a closed-twin-free graph $G$ is a dominating 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 that there exists an absolute constant $c$ such that for every connected graph $G$ of order $n$ and maximum degree $Δ$, the graph $G$ admits an identifying code of size at most $( \frac{Δ-1}Δ )n +c$. We provide significant support for this conjecture by exactly characterizing every tree requiring a positive constant $c$ together with the exact value of the constant. Hence, proving the conjecture for trees. For $Δ=2$ (the graph is a path or a cycle), it is long known that $c=3/2$ suffices. For trees, for each $Δ\ge 3$, we show that $c=1/Δ\le 1/3$ suffices and that $c$ is required to have a positive value only for a finite number of trees. In particular, for $Δ= 3$, there are 12 trees with a positive constant $c$ and, for each $Δ\ge 4$, the only tree with positive constant $c$ is the $Δ$-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]. We remark that there are infinitely many trees for which the bound is tight when $Δ=3$; for every $Δ\ge 4$, we construct an infinite family of trees of order $n$ with identification number very close to the bound, namely $\left( \frac{Δ-1+\frac{1}{Δ-2}}{Δ+\frac{2}{Δ-2}} \right) n > (\frac{Δ-1}Δ ) n -\frac{n}{Δ^2}$. Furthermore, we also give a new tight upper bound for identification number on trees by showing that the sum of the domination and identification numbers of any tree $T$ is at most its number of vertices. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_13172 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Identifying codes in graphs of given maximum degree: Characterizing trees Chakraborty, Dipayan Foucaud, Florent Henning, Michael A. Lehtilä, Tuomo Combinatorics 05C69, 05C05, 05C07 An identifying code of a closed-twin-free graph $G$ is a dominating 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 that there exists an absolute constant $c$ such that for every connected graph $G$ of order $n$ and maximum degree $Δ$, the graph $G$ admits an identifying code of size at most $( \frac{Δ-1}Δ )n +c$. We provide significant support for this conjecture by exactly characterizing every tree requiring a positive constant $c$ together with the exact value of the constant. Hence, proving the conjecture for trees. For $Δ=2$ (the graph is a path or a cycle), it is long known that $c=3/2$ suffices. For trees, for each $Δ\ge 3$, we show that $c=1/Δ\le 1/3$ suffices and that $c$ is required to have a positive value only for a finite number of trees. In particular, for $Δ= 3$, there are 12 trees with a positive constant $c$ and, for each $Δ\ge 4$, the only tree with positive constant $c$ is the $Δ$-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]. We remark that there are infinitely many trees for which the bound is tight when $Δ=3$; for every $Δ\ge 4$, we construct an infinite family of trees of order $n$ with identification number very close to the bound, namely $\left( \frac{Δ-1+\frac{1}{Δ-2}}{Δ+\frac{2}{Δ-2}} \right) n > (\frac{Δ-1}Δ ) n -\frac{n}{Δ^2}$. Furthermore, we also give a new tight upper bound for identification number on trees by showing that the sum of the domination and identification numbers of any tree $T$ is at most its number of vertices. |
| title | Identifying codes in graphs of given maximum degree: Characterizing trees |
| topic | Combinatorics 05C69, 05C05, 05C07 |
| url | https://arxiv.org/abs/2403.13172 |