Forcibly unicyclic and bicyclic graphic sequences
Peiyi Duan, Yingzhi Tian
TL;DR
This work addresses the problem of identifying graphic sequences whose every realization is unicyclic or bicyclic. It advances a switching-based, cycle-decomposition approach to constrain the possible realizations and then provides complete classifications: for forcibly unicyclic sequences, the allowed forms are $D=(2^5)$, $D=(3,2^4,1)$, or the families $D=(n-2,2^3,1^{n-4})$ and $D=(r,s,t,1^{n-3})$ with $r\ge s\ge t\ge2$ and $r+s+t=n+3$ (plus small-$n$ cases). For forcibly bicyclic sequences, it gives explicit lists of small-n sequences and larger families $D=(n-1,2^4,1^{n-5})$, $D=(n-2,2^5,1^{n-6})$, $D=(n-2,3,2^3,1^{n-5})$, and $D=(r,s,t,2,1^{n-4})$, together with structural lemmas that bound degrees and leverage switchings to force specific subgraph configurations. The results extend prior work on forcible acyclicity by delivering a complete, explicit description of degree sequences that force unicyclic or bicyclic realizations, with implications for graph realization problems under fixed degree sequences. Throughout, all arguments rely on the interplay between cycle structure and degree constraints via $\sum_{i=1}^n d_i=2n+O(1)$ relationships and switching operations.
Abstract
A sequence $D=(d_1,d_2,\ldots,d_n)$ of non-negative integers is called a graphic sequence if there is a simple graph with vertices $v_1,v_2,\ldots,v_n$ such that the degree of $v_i$ is $d_i$ for $1\leq i\leq n$. Given a graph theoretical property $\mathcal{P}$, a graphic sequence $D$ is forcibly $\mathcal{P}$ graphic if each graph with degree sequence $D$ has property $\mathcal{P}$. A graph is acyclic if it contains no cycles. A connected acyclic graph is just a tree and has $n-1$ edges. A graph of order $n$ is unicyclic (resp. bicyclic) if it is connected and has $n$ (resp. $n+1$) edges. Bar-Noy, Böhnlein, Peleg and Rawitz [Discrete Mathematics 346 (2023) 113460] characterized forcibly acyclic and forcibly connected acyclic graphic sequences. In this paper, we aim to characterize forcibly unicyclic and forcibly bicyclic graphic sequences.