A sequence D= (d₁, d₂, , dₙ) of non-negative integers is called a graphic sequence if there is a simple graph with vertices v₁, v₂, , vₙ such that the degree of vᵢ is dᵢ for 1 i n. Given a graph theoretical property P, a graphic sequence D is forcibly P graphic if each graph with degree sequence D has property 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\"ohnlein, 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.
Duan et al. (Tue,) studied this question.
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