Realizing Graphs with Cut Constraints
Lucas de Oliveira Silva, V\'itor Gomes Chagas, Samuel Pla\c{c}a de, Paula, Greis Yvet Oropeza Quesqu\'en, U\'everton dos Santos Souza

TL;DR
This paper extends the classical graph realization problem by incorporating cut constraints, providing polynomial-time solutions for small sets and proving NP-hardness for larger sets, highlighting the problem's computational complexity.
Contribution
It introduces a generalized graph realization problem with cut constraints and characterizes its complexity, offering polynomial algorithms for small sets and NP-hardness results for larger sets.
Findings
Polynomial-time solvable when each set has size at most three.
NP-hardness proven when sets of size four are included.
Hardness holds even when all degrees are one.
Abstract
Given a finite non-decreasing sequence of natural numbers, the Graph Realization problem asks whether is a graphic sequence, i.e., there exists a labeled simple graph such that is the degree sequence of this graph. Such a problem can be solved in polynomial time due to the Erd\H{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 of natural numbers (representing the trivial edge cut sizes) and a list of nontrivial cut constraints composed of pairs where , and is a natural number. In such a problem, we are asked whether there is a simple graph with vertex set such that has…
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Taxonomy
TopicsConstraint Satisfaction and Optimization · Model-Driven Software Engineering Techniques · Advanced Database Systems and Queries
