关于边围长序列的可实现性

中文摘要 AI 辅助

简单连通图中边\(e\)的边围长是包含\(e\)的最短圈的长度，若不存在这样的圈则\(g_e = \infty\)。图的边围长序列是其所有边的边围长值的非递减序列。我们证明序列\(S\)可作为简单连通图的边围长序列当且仅当它满足一个递归准则：将\(S = S_0 \uplus (g^{(m)})\)，其中\(g\)是\(S\)中最大边围长值且重数为\(m\)，\(S_0\)是前缀子序列，\(S\)可实现当且仅当\(S_0\)可实现且重数\(m\)在由\(g\)和实现\(S_0\)的图可达到的最大直径\(d^*_{S_0}\)完全确定的集合中。我们还确定了任何可实现序列的\(d^*_S\)：对于常数序列\((g^{(m)})\)，当\(g\)为偶数时得到一个闭式公式，当\(g\)为奇数时得到一个递归公式。对于一般序列，我们提供了一个计算\(d^*_S\)的递归算法以及直径达到的图的显式构造。

英文摘要

The edge-girth of an edge $e$ in a simple connected graph is the length of a shortest cycle containing $e$, with $g_e = \infty$ if no such cycle exists, and the edge-girth sequence of a graph is the nondecreasing sequence of the edge-girths of its edges. We characterize the sequences that are realizable as the edge-girth sequence of a simple connected graph. A constant sequence $(g^{(m)})$ with $g$ finite and $m \geq 1$ is realizable if and only if $m = g$ or $m \geq \lceil 3g/2 \rceil$. Otherwise, writing $S = S_0 \uplus (g^{(m)})$, where $g$ is the maximum value of $S$ and $m$ its multiplicity, we prove that $S$ is realizable if and only if $S_0$ is realizable and $m$ lies in an explicit set determined by $g$ and by the maximum diameters of the graphs realizing $S_0$, alone or extended by fewer than $g$ edges of edge-girth $g$. The proof rests on a chaining lemma, which turns a shortest cycle of a realization into a lower bound on the maximum diameter of the realizations of a shorter sequence.

发表机构

• Télécom Paris, Institut Polytechnique de Paris（巴黎电信学院，巴黎理工学院）
• LTCI, Inria, Télécom Paris, Institut Polytechnique de Paris（通信与网络实验室，法国国家信息与自动化研究所，巴黎电信学院，巴黎理工学院）

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