Realizing Graphs with Cut Constraints

TL;DR

This work defines Graph Realization with Cut Constraints (GR-C), a natural generalization of the graph realization problem that adds nontrivial edge-cut constraints to a prescribed degree sequence. The authors establish a clear complexity dichotomy: GR-C is solvable in polynomial time when the largest cut size satisfies $w(\mathcal{L})\le 3$ (with $w=2$ reducing to an $f$-factor problem and $w=3$ reducible to a $w\le 2$ instance); however, the problem becomes NP-hard for $w(\mathcal{L})\ge 4$, even when the degree sequence consists of ones. They further show NP-completeness for $w(\mathcal{L})=6$ under restricted, bipartite, subcubic possibility graphs, while GR-C remains tractable when the possibility graph is a tree. The hardness results are supported by reductions from $1$-in-$3$-SAT$_{(2,1)}$ and 3DM-3, illustrating the tight dependence on cut size and graph structure. These results connect classic $f$-factor theory with cut-based constraints and open avenues for studying GR-C in restricted graph classes and alternative logical reductions.

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 Erdős 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 $S_j\subset \{v_1,\ldots,v_n\}$, 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 $\neq$ 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 $1$.

Figures (4)

• Figure 1: Illustration of all cases for a cut $(S, \ell)$ with $S = \{ u, v, w \}$, assuming $d_u = d_v = d_w = 2$ (so $d(S) = 6$). Solid edges represent possible edges, dashed edges are forbidden, and blue-highlighted edges belong to a realization. In each case, the left image shows a realization satisfying $(S, \ell)$, while the right image shows the equivalent realization of the modified instance without the cut.
• Figure 2: Illustration of the possibility graph $\mathcal{G}$ built from an instance $(X, \phi, k)$ of $k$-True$1$-in-$3$-SAT$_{(2,1)}$ with $X = \{ x_1, x_2, x_3, x_4 \}$, $\phi = (\bar{x}_1 \mathcal{+} x_3)(x_1 \mathcal{+} x_2 \mathcal{+} x_4)(x_1 \mathcal{+} \bar{x}_4)(\bar{x}_2 \mathcal{+} \bar{x}_3)(x_2 \mathcal{+} x_3 \mathcal{+} x_4)$ and $k = 1$. Gray vertices represent artificial vertices created for clauses with only two literals. The highlighted edges show an example of a feasible realization for such an instance.
• Figure 3: A 3DM-3 instance example where $T=\{(x_1, y_1, z_1), (x_1, y_2, z_2),\\ (x_2, y_1, z_1), (x_2, y_2, z_1), (x_3, y_2, z_2), (x_3, y_3, z_3)\}$. Distinct edge types are assigned to each triple, with a solution highlighted. The right image depicts the possibility graph of the reduced GR-C instance, with a feasible realization highlighted.
• Figure :

Theorems & Definitions (16)

• theorem thmcountertheorem: Erdős and Gallai erdos60
• remark thmcounterremark
• remark thmcounterremark
• lemma thmcounterlemma
• proof
• theorem thmcountertheorem
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma