Counting perfect edge dominating sets: extremal results and linear-time algorithms

Abstract: An edge of a graph \emph{dominates} itself and each edge adjacent to it. A \emph{perfect edge dominating set} is a subset of edges such that each edge outside the subset is dominated by exactly one edge of the subset. In this article, we characterize the extremal graphs on $n$ vertices in the classes of trees, forests, and chordal graphs with respect to the number of perfect edge dominating sets. Moreover, we derive linear-time algorithms for counting perfect edge dominating sets and for counting dominating induced matchings in generalized series-parallel graphs and chordal graphs.

• The paper introduces a novel enumeration framework that characterizes extremal graphs with maximized perfect edge dominating sets.
• It establishes recurrences and explicit bounds for trees, forests, and chordal graphs, revealing paths and triangles as extremal structures.
• The authors design dynamic programming algorithms that count PED-sets and induced matchings in O(n+m) time in structured graph classes.

Counting Perfect Edge Dominating Sets: Extremal Graphs and Linear-Time Algorithms

Introduction and Problem Context

The enumeration and algorithmic study of perfect edge dominating sets (PED-sets) represent a substantial yet underexplored area in combinatorial optimization and graph theory. While classical vertex domination and matching problems have been deeply analyzed from both enumeration and algorithmic perspectives, analogous questions in the edge domain exhibit distinct structural and computational properties. This paper undertakes a thorough investigation into the enumeration of PED-sets, provides tight extremal results over several fundamental graph classes, and establishes efficient linear-time algorithms for counting both PED-sets and dominating induced matchings (DIMs) in broad classes of graphs.

A PED-set in a graph $G$ is a subset of edges such that every edge outside the subset is dominated by exactly one member of the subset, where domination is defined by adjacency or equality. These sets generalize the notion of efficient edge domination and induce a bijection with certain 3-colorings of the vertex set. The problem of counting PED-sets, and by extension, DIMs (PED-sets that correspond to induced matchings), is intractable for general graphs, but there is a lack of structural results and efficient algorithms in restricted classes. This work closes important gaps in both enumeration and complexity for trees, forests, chordal graphs, and generalized series-parallel graphs.

Theoretical Advances: Extremal Enumeration Results

Trees and Forests

A key result is the complete tight characterization of extremal trees and forests regarding the number of PED-sets:

• Paths as Extremal Trees: For any $n$, the $n$-vertex path $P_n$ maximizes the number of PED-sets among all $n$-vertex trees, except for $n=4,5$, where the star $K_{1,3}$ and $K_{1,4}$ are also extremal. The enumeration for paths is shown to follow the recurrence:

$$\tilde{\mu}(P_n) = \tilde{\mu}(P_{n-1}) + \tilde{\mu}(P_{n-3}), \quad n \geq 4,$$

with explicit initial values.

• Extremal Forests: For forests, the extremal configuration depends on $n$ and is a unique path $n$0 for sufficiently large $n$1 (specifically, $n$2). For smaller $n$3, the extremal family includes certain disjoint unions of smaller paths.

Chordal Graphs

• Decomposition and Characterization: For chordal graphs, the extremal graphs with respect to PED-sets are exactly those that are disjoint unions of triangles and extremal forests, capturing the inherent decomposability of chordal graphs via their clique trees.
• Enumeration Formula: The maximal number of PED-sets in an $n$4-vertex chordal graph, $n$5, is given explicitly as:

$n$6

The extremal graphs achieving $n$7 are exactly those comprised of the appropriate number of triangles and an extremal forest, with all possible combinations enumerated.

• Structural Rigidity: A theorem establishes that any extremal (chordal or general) graph must contain every triangle as a separate connected component, excluding the possibility of triangles contributing in larger, more complex substructures.

Cycles and Conjectural Bound

The paper conjectures (with supporting partial results) that among all connected $n$8-vertex graphs, the $n$9-cycle $n$0 maximizes the number of PED-sets, with $n$1 satisfying the same recurrence as for paths, but with provably larger initial values.

Linear-Time Algorithms: Dynamic Programming for PED-Counting

Generalized Series-Parallel (GSP) and Chordal Graphs

The authors design dynamic programming algorithms, inspired by prior structural decompositions [Lu, Ko, Tang 2002], that count PED-sets and DIMs in linear time for GSP and chordal graphs. Key aspects include:

• Graph Decomposition: GSP graphs admit parse trees corresponding to their recursive composition, facilitating stateful bottom-up dynamic programming. Chordal graphs, via clique trees, can be reduced by extracting and separating large cliques (which are forced to be monochromatically black), reducing the problem to GSP graphs.
• State Definition: States in the DP maintain, for each pair of distinguished (terminal) vertices, the color assignments (white, yellow of various degrees, black), allowing composition through series and parallel constructs.
• Algorithmic Complexity: Both PED- and DIM-counting algorithms run in $n$2 time, as each composition step manipulates a bounded set of states and each edge or composition is processed once.
• Generalization: By adjusting the state space (e.g., forbidding black colors), the same technique delivers efficient algorithms for related edge domination and matching structures.

Numerical Results and Explicit Bounds

The recurrence-based enumeration leads to explicit numeric upper bounds for PED-set counts in the considered graph classes, which are sharp and achieved exactly by the characterized extremal graphs. For example:

Graph Class	Maximum $n$3 for $n$4 vertices
Trees	$n$5 (by recurrence)
Forests	$n$6 for $n$7
Chordal	$n$8 as detailed above

These results resolve the extremal enumeration problem for PED-sets for trees, forests, and chordal graphs.

Implications and Directions for Future Research

Practical Implications

The linear-time algorithms enable efficient enumeration of PED-sets and DIMs in large-scale applications, making possible combinatorial enumeration, probabilistic sampling, or backbone extraction for edge-based structures in networks modeled by GSP or chordal graphs. The extremal characterizations provide structural insight relevant for benchmarking and testing combinatorial algorithms.

Theoretical Implications and Open Problems

• Broader Classes & Complexity: The approach strongly exploits the decomposition structure of GSP and chordal graphs. Extending these enumeration results and efficient algorithms to broader classes (e.g., planar or $n$9-free graphs for larger $P_n$0) remains challenging due to the loss of such recursive tractability.
• Conjectures for General Graphs: The conjecture that cycles maximize PED-sets among connected graphs is not resolved for all $P_n$1; progress here would require new tools for controlling PED-set multiplicity under graph gluing and cycle expansion.
• Synthesis and Enumeration: The explicit bijection between PED-sets and valid 3-colorings suggests potential for efficient uniform random sampling, which could be explored further for approximate counting or sampling applications.
• Algorithm Engineering: The dynamic programming formalism may be adapted to hybrid settings, e.g., using tree decompositions in graphs of bounded treewidth or in parameterized complexity frameworks.

Conclusion

The paper delivers a comprehensive resolution of the extremal enumeration and algorithmic aspects of perfect edge dominating sets in trees, forests, and chordal graphs. The contributions include tight extremal structure theorems, explicit recurrences and bounds, and practical $P_n$2 algorithms for enumerating PED-sets and DIMs in structured graph classes. These advances not only refine the understanding of edge-based domination in graphs but also provide effective tools and explicit benchmarks for future research in both theoretical and applied domains of combinatorics and algorithms.