Enumerating minimal dominating sets in the (in)comparability graphs of bounded dimension posets

TL;DR

This work studies the enumeration of inclusion-wise minimal dominating sets across two graph families tied to posets: incomparability graphs and their complements, comparability graphs. It delivers an output-polynomial, polynomial-space algorithm for incomparability graphs of posets with bounded dimension using a flashlight-search framework guided by a geometric representation, and an incremental-polynomial, initially exponential-space approach for comparability graphs via the flipping method, later improved to polynomial space. Central to the approach is reducing flipping to red-blue domination in posets, enabling tractable enumeration under bounded-dimension and specific structural restrictions (e.g., S_t-free posets) through conformality and dualization techniques. The paper also clarifies how the incomparability setting admits a polynomial-delay algorithm leveraging a curved-visibility representation of the dimension-bounded poset, yielding practical algorithms and advancing the state-of-the-art for several graph classes. Overall, the results illuminate how dimension, poset structure, and specialized domination variants interact to ease Dom-Enum in otherwise hard instances, while outlining open challenges for dimension-free incomparability graphs.

Abstract

Enumerating minimal transversals in a hypergraph is a notoriously hard problem. It can be reduced to enumerating minimal dominating sets in a graph, in fact even to enumerating minimal dominating sets in an incomparability graph. We provide an output-polynomial time algorithm for incomparability graphs whose underlying posets have bounded dimension. Through a different proof technique, we also provide an output-polynomial algorithm for their complements, i.e., for comparability graphs of bounded dimension posets. Our algorithm for incomparability graphs is based on flashlight search and relies on the geometrical representation of incomparability graphs with bounded dimension, as given by Golumbic et al. in 1983. It runs with polynomial delay and only needs polynomial space. Our algorithm for comparability graphs is based on the flipping method introduced by Golovach et al. in 2015. It performs in incremental-polynomial time and requires exponential space. In addition, we show how to improve the flipping method so that it requires only polynomial space. Since the flipping method is a key tool for the best known algorithms enumerating minimal dominating sets in a number of graph classes, this yields direct improvements on the state of the art.

Figures (5)

• Figure 1: The Hasse diagram of a poset (left), its comparability graph (middle), and incomparability graph (right).
• Figure 2: The standard example of order four.
• Figure 3: An illustration of the flipping operation on a dominating set $D$ such that $G[D]$ contains at least one edge incident to some vertex $u$, here depicted by $uw$. Black vertices are elements of $D$, white vertices are some elements of $\mathop{\mathrm{Priv}}\nolimits(D,u)$. Dashed discs represent closed neighborhoods, and plain disks represent private neighborhoods.
• Figure 4: A minimal dominating set $D$ (on the left) and its parent $D^*=\mathop{\mathrm{Parent}}\nolimits_{uv}(D)$ (on the right) represented by black vertices in the Hasse diagram of the underlying poset of a comparability graph. The edge induced by $D$ and incident to $u$ is depicted by $uw$.
• Figure 5: A poset $P$ and its incomparability graph $G$ as an intersection graph of curves induced by four linear extensions witnessing the dimension.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Proposition 3.1: golovach2015flipping
• Lemma 3.2: golovach2015flipping
• Lemma 3.3
• proof
• Corollary 3.4
• Lemma 4.1
• proof
• Claim 4.2