On the Realizability of Edge-Girth Sequences

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. The edge-girth sequence of a graph is the nondecreasing sequence of edge-girth values over all its edges. We prove that a sequence $S$ is realizable as the edge-girth sequence of a simple connected graph if and only if it satisfies a recursive criterion: writing $S = S_0 \uplus (g^{(m)})$ where $g$ is the maximum edge-girth value of $S$ with multiplicity $m$ and $S_0$ is the prefix subsequence, $S$ is realizable if and only if $S_0$ is realizable and the multiplicity $m$ lies in a set entirely determined by $g$ and the maximum diameter $d^*_{S_0}$ achievable by graphs realizing $S_0$. We further determine $d^*_S$ for any realizable sequence: for constant sequences $(g^{(m)})$, we obtain a closed-form formula when $g$ is even and a recursive formula when $g$ is odd. For general sequences, we provide a recursive algorithm computing $d^*_S$ together with explicit constructions of diameter-achieving graphs.