On constrained intersection representations of graphs and digraphs

TL;DR

This paper advances the study of directed intersection representations (DIN) for DAGs by proving a polynomial-time algorithm for triangle-free Hamiltonian DAGs, expressing the optimal DIN via a simple formula din(D) = |A| + b(V) − ν(G,b) and reducing the problem to a maximum weight b-matching. The approach builds a chain of reductions among directed, weak, and poset-based intersection representations, leveraging the α-ranking and ell-constrained representations on triangle-free graphs to achieve exact results. It also links DIN to classic intersection-number frameworks on undirected graphs and expands the landscape with variants like ell-constrained IN and weak DIN, mapping out relationships and time complexities. The work further provides specialized insights for bipartite cases, discusses constant-factor approximations for broader DAG classes, and outlines open questions on the limits of these representations and their computational boundaries, offering a rich roadmap for both theory and potential applications in constrained resource scheduling and graph representation.

Abstract

We study the problem of determining optimal directed intersection representations of DAGs in a model introduced by Kostochka, Liu, Machado, and Milenkovic [ISIT2019]: vertices are assigned color sets so that there is an arc from a vertex $u$ to a vertex $v$ if and only if their color sets have nonempty intersection and $v$ gets assigned strictly more colors than $u$, and the goal is to minimize the total number of colors. We show that the problem is polynomially solvable in the class of triangle-free and Hamiltonian DAGs and also disclose the relationship of this problem with several other models of intersection representations of graphs and digraphs.

Figures (3)

• Figure 1: The relationships among the different types of intersection representations considered in this paper. The arc $\mathit{Prob}(a) \rightarrow \mathit{Prob}(b)$ is to be read "$\mathit{Prob}(a)$ is a special case of $\mathit{Prob}(b)$." A label on an arc specifies the restriction on the class of instances for which the relation is proved to hold, along with a reference to the statement justifying the relation. The highlighted parameters are introduced in this paper.
• Figure 2: An example of application of \ref{['triangle-free-Hamiltonian-DIN-main']} on a digraph $D = (V,A)$. The figure shows the underlying graph $G=(V,E)$; the digraph $D$ is obtained by orienting all the edges of $G$ from the left to the right. We explicitly give the parameters used in \ref{['triangle-free-Hamiltonian-DIN-main']}: degrees, capacity function $b()$, and edge values of an optimal $b$-matching (we only display the non-zero values). The fact that the $b$-matching is indeed optimal can either be verified by observing that its total weight, $40$, matches the upper bound given by the $b$-weight of some vertex cover of $G$ (consider, e.g., $\{v_1,v_3,v_5,v_7,v_9,v_{11}\}$), or by using the fact that $\nu(G,b) = |A| + b(V) - \textsf{din}(D) = 40$, since $\textsf{din}(D) = 77$ (as shown in MR4231959), $|A| = 21$, and $b(V) = 96$.
• Figure 3: Steps of the proof of \ref{['triangle-free-Hamiltonian-DIN-main']}.

Theorems & Definitions (43)

• Theorem 1
• Proposition 2
• proof
• Proposition 3
• proof
• Theorem 5
• Lemma 5
• Lemma 5
• Theorem 6
• Theorem 7