Forcibly unicyclic and bicyclic graphic sequences

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, B\"{o}hnlein, 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.