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Efficient Enumeration of Dominating Sets for Sparse Graphs
Data Structures and Algorithms · Submitted 22 Feb 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this paper, we address enumeration problems for dominating sets from sparse graphs which are degenerate graphs and graphs with large girth, and we propose two algorithms for solving the problems.
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The abstract, sentence by sentence
A dominating set $D$ of a graph $G$ is a set of vertices such that any vertex in $G$ is in $D$ or its neighbor is in $D$. Enumeration of minimal dominating sets in a graph is one of central problems in enumeration study since enumeration of minimal dominating sets corresponds to enumeration of minimal hypergraph transversal. However, enumeration of dominating sets including non-minimal ones has not been received much attention. In this paper, we address enumeration problems for dominating sets from sparse graphs which are degenerate graphs and graphs with large girth, and we propose two algorithms for solving the problems. The first algorithm enumerates all the dominating sets for a $k$-degenerate graph in $O(k)$ time per solution using $O(n + m)$ space, where $n$ and $m$ are respectively the number of vertices and edges in an input graph. That is, the algorithm is optimal for graphs with constant degeneracy such as trees, planar graphs, $H$-minor free graphs with some fixed $H$. The second algorithm enumerates all the dominating sets in constant time per solution for input graphs with girth at least nine.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A dominating set $D$ of a graph $G$ is a set of vertices such that any vertex in $G$ is in $D$ or its neighbor is in $D$. | background 1.00 | 0.04 | 0.02 | asserted 1.00 |
| s1 | Enumeration of minimal dominating sets in a graph is one of central problems in enumeration study since enumeration of minimal dominating sets corresponds to enumeration of minimal hypergraph transversal. | background 1.00 | 0.04 | 0.04 | asserted 1.00 |
| s2 | However, enumeration of dominating sets including non-minimal ones has not been received much attention. | gap 1.00 | 0.05 | 0.03 | asserted 0.99 |
| s3 | In this paper, we address enumeration problems for dominating sets from sparse graphs which are degenerate graphs and graphs with large girth, and we propose two algorithms for solving the problems. | contribution 1.00 | 0.91 | 0.21 | asserted 1.00 |
| s4 | The first algorithm enumerates all the dominating sets for a $k$-degenerate graph in $O(k)$ time per solution using $O(n + m)$ space, where $n$ and $m$ are respectively the number of vertices and edges in an input graph. | result 0.46 | 0.94 | 0.37 | asserted 1.00 |
| s5 | That is, the algorithm is optimal for graphs with constant degeneracy such as trees, planar graphs, $H$-minor free graphs with some fixed $H$. | interpretation 0.95 | 0.69 | 0.13 | asserted 1.00 |
| s6 | The second algorithm enumerates all the dominating sets in constant time per solution for input graphs with girth at least nine. | contribution 0.48 | 0.93 | 0.20 | asserted 1.00 |
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Original abstract
A dominating set $D$ of a graph $G$ is a set of vertices such that any vertex in $G$ is in $D$ or its neighbor is in $D$. Enumeration of minimal dominating sets in a graph is one of central problems in enumeration study since enumeration of minimal dominating sets corresponds to enumeration of minimal hypergraph transversal. However, enumeration of dominating sets including non-minimal ones has not been received much attention. In this paper, we address enumeration problems for dominating sets from sparse graphs which are degenerate graphs and graphs with large girth, and we propose two algorithms for solving the problems. The first algorithm enumerates all the dominating sets for a $k$-degenerate graph in $O(k)$ time per solution using $O(n + m)$ space, where $n$ and $m$ are respectively the number of vertices and edges in an input graph. That is, the algorithm is optimal for graphs with constant degeneracy such as trees, planar graphs, $H$-minor free graphs with some fixed $H$. The second algorithm enumerates all the dominating sets in constant time per solution for input graphs with girth at least nine.
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