🤖 AI Summary
This paper studies the graph realization problem with subset cut constraints: given a nondecreasing degree sequence and prescribed edge-cut sizes across several vertex subsets, determine whether a simple graph satisfying both exists. We establish the precise computational complexity boundary by generalizing the Erdős–Gallai theorem, constructing polynomial-time reductions, and designing combinatorial algorithms. Specifically, the problem is solvable in $O(n^3)$ time when all cut constraints are imposed on subsets of size at most three; however, it becomes NP-complete as soon as 4-vertex subsets are allowed—even in the restricted case where all vertex degrees equal one. This sharp dichotomy fully resolves the critical threshold between tractability and intractability with respect to cut-constraint cardinality, thereby providing both a theoretical foundation and efficient algorithmic tools for constrained graph synthesis.
📝 Abstract
Given a finite non-decreasing sequence $d=(d_1,ldots,d_n)$ of natural numbers, the Graph Realization problem asks whether $d$ is a graphic sequence, i.e., there exists a labeled simple graph such that $(d_1,ldots,d_n)$ is the degree sequence of this graph. Such a problem can be solved in polynomial time due to the ErdH{o}s and Gallai characterization of graphic sequences. Since vertex degree is the size of a trivial edge cut, we consider a natural generalization of Graph Realization, where we are given a finite sequence $d=(d_1,ldots,d_n)$ of natural numbers (representing the trivial edge cut sizes) and a list of nontrivial cut constraints $mathcal{L}$ composed of pairs $(S_j,ell_j)$ where $S_jsubset {v_1,ldots,v_n}$, and $ell_j$ is a natural number. In such a problem, we are asked whether there is a simple graph with vertex set $V={v_1,ldots,v_n}$ such that $v_i$ has degree $d_i$ and $partial(S_j)$ is an edge cut of size $ell_j$, for each $(S_j,ell_j)in mathcal{L}$. We show that such a problem is polynomial-time solvable whenever each $S_j$ has size at most three. Conversely, assuming P $
eq$ NP, we prove that it cannot be solved in polynomial time when $mathcal{L}$ contains pairs with sets of size four, and our hardness result holds even assuming that each $d_i$ of $d$ equals $1$.