Perfect Edge Dominating Set

• Perfect edge dominating sets are defined as subsets of a graph's edges that uniquely dominate each non-selected edge, serving as an intermediate concept between standard edge domination and DIMs.
• Structural characterizations via vertex 3-colorings enable precise analyses and practical algorithms for classes like P5-free, circular-arc, and chordal graphs.
• Hardness proofs and counting methods highlight a rich complexity landscape, including NP-completeness for claw-free and bounded-degree graphs alongside efficient algorithms for specific classes.

A perfect edge dominating set of a graph $G$ is a set $P\subseteq E(G)$ such that every edge in $E(G)\setminus P$ is dominated by exactly one edge of $P$, where an edge dominates itself and every adjacent edge. Equivalently, each edge outside $P$ has a unique neighbor in $P$ in the line graph (Grippo et al., 4 Jul 2026). Perfect edge domination lies between ordinary edge domination and efficient edge domination: an edge dominating set only requires domination by at least one selected edge, whereas an efficient edge dominating set—also called a dominating induced matching (DIM)—requires that every edge of the graph be dominated by exactly one selected edge and that the selected edges be pairwise non-adjacent (Lin et al., 2017, Grippo et al., 4 Sep 2025). The subject combines structural graph theory, hardness reductions, hereditary-class dichotomies, and exact algorithms, with particularly strong results on $P_5$-free graphs, $P_6$-free graphs, circular-arc graphs, generalized series-parallel graphs, and chordal graphs (Lin et al., 2017, Grippo et al., 4 Sep 2025, Lin et al., 2015, Grippo et al., 4 Jul 2026).

1. Definitions and basic variants

For an undirected graph $G$, an edge $e$ dominates itself and every edge adjacent to it. A set $P\subseteq E(G)$0 is an edge dominating set if every edge of $P\subseteq E(G)$1 is dominated by some edge in $P\subseteq E(G)$2. It is a perfect edge dominating set if every edge in $P\subseteq E(G)$3 is dominated by exactly one edge of $P\subseteq E(G)$4 (Lin et al., 2017). The corresponding decision problem is:

• PERFECT EDGE DOMINATION INPUT: Graph $P\subseteq E(G)$5, integer $P\subseteq E(G)$6 QUESTION: Does $P\subseteq E(G)$7 contain a perfect edge dominating set $P\subseteq E(G)$8 of size $P\subseteq E(G)$9? (Lin et al., 2017)

The optimization version asks for a least-cardinality perfect edge dominating set. The weighted version asks for a minimum-weight perfect edge dominating set, where edge weights may be arbitrary reals, even negative (Lin et al., 2017).

Two special subclasses are standard. A trivial perfect edge dominating set is $E(G)\setminus P$0, equivalently $E(G)\setminus P$1 in the coloring formalism below. A proper perfect edge dominating set is one that is neither trivial nor a DIM; for connected graphs with at least three vertices, this is the case $E(G)\setminus P$2, $E(G)\setminus P$3, and $E(G)\setminus P$4 (Lin et al., 2017, Grippo et al., 4 Sep 2025).

Efficient edge domination is strictly stronger. A DIM is a PED-set in which no two selected edges are adjacent, and every edge of $E(G)\setminus P$5 is dominated by exactly one selected edge. Every DIM is a PED-set, and whenever a DIM exists it is a PED-set of minimum cardinality (Grippo et al., 4 Jul 2026). This distinction is central in both hardness proofs and positive algorithms: many tractable cases first test for a DIM, then compare it with the trivial set $E(G)\setminus P$6, and finally analyze the genuinely proper PED configurations (Lin et al., 2017, Lin et al., 2015, Grippo et al., 4 Sep 2025).

2. Structural characterization by vertex colorings

A central structural tool is the vertex $E(G)\setminus P$7-coloring associated with a PED-set. Given a connected graph and a perfect edge dominating set $E(G)\setminus P$8, one partitions the vertices into:

• black vertices $E(G)\setminus P$9: vertices incident with at least two edges of $P$0,
• yellow vertices $P$1: vertices incident with exactly one edge of $P$2,
• white vertices $P$3: vertices incident with no edge of $P$4 (Lin et al., 2017, Grippo et al., 4 Jul 2026).

In the circular-arc treatment, the same role is played by a black/gray/white coloring, with $P$5 the set of vertices incident to selected edges, $P$6, $P$7, and $P$8 (Lin et al., 2015).

The later PED literature isolates the valid colorings precisely. A coloring is valid exactly when:

1. $P$9 is an independent set;
2. a vertex is yellow iff it is a leaf of $P$0;
3. every white vertex has only yellow neighbors; and
4. every black vertex has no white neighbors and degree at least two (Grippo et al., 4 Jul 2026).

Conversely, every coloring satisfying these conditions corresponds to a unique PED-set, namely the set of edges whose endpoints are both non-white (Grippo et al., 4 Jul 2026). The earlier circular-arc formulation encodes the same mechanism through two properties: (P1) each gray vertex has exactly one non-white neighbor, all its other neighbors being white; (P2) if $P$1, then $P$2, hence $P$3 is independent (Lin et al., 2015).

This coloring formalism has strong immediate consequences. Every induced $P$4 with $P$5 must have all vertices black. Every induced triangle has either three black vertices or two yellow vertices and one white vertex (Lin et al., 2017). These local constraints are repeatedly used to rule out proper PED-sets in triangle-rich graph classes and to force finite pattern lists in $P$6-free and $P$7-free algorithms (Lin et al., 2017, Grippo et al., 4 Sep 2025).

For weighted problems, the coloring viewpoint yields a useful objective transformation. If $P$8 and

$P$9

then for the PED-set induced by a valid coloring $P$0,

$P$1

Minimizing $P$2 is therefore equivalent to maximizing $P$3 (Grippo et al., 4 Sep 2025).

3. Complexity landscape

The general problem is computationally hard in several strong senses. One NP-completeness result applies to claw-free graphs of degree at most $P$4. The reduction is based on the shield graph and a magnification operation. If $P$5 is cubic and $P$6 is its magnification, then

$P$7

which yields NP-hardness for minimum perfect edge domination on claw-free graphs of degree at most $P$8 (Lin et al., 2017).

A second hardness construction applies to bounded-degree graphs with large girth. For an $P$9-regular graph $P_5$0, the $P_5$1-subdivision $P_5$2 forces rigid local behavior: for every perfect edge dominating set $P_5$3 of $P_5$4, each subdivided edge gadget contains at least $P_5$5 selected edges, and the equality case induces a specific white/yellow endpoint pattern (Lin et al., 2017). This leads to NP-hardness of the cardinality problem on graphs of bounded degree $P_5$6 and girth at least $P_5$7, for any fixed $P_5$8, and in particular on $P_5$9-regular graphs for every $P_6$0 (Lin et al., 2017).

A major hereditary-class theorem gives a sharp bounded-degree dichotomy. Let $P_6$1 be fixed, and let $P_6$2 be the class of $P_6$3-free graphs of maximum degree at most $P_6$4, for fixed $P_6$5. Then perfect edge domination is polynomial-time solvable if $P_6$6 is a linear forest, and NP-complete otherwise (Lin et al., 2017). The hard side splits according to whether $P_6$7 contains an induced cycle or a vertex of degree at least $P_6$8; the easy side uses the fact that bounded-degree $P_6$9-free connected graphs have bounded size (Lin et al., 2017).

Later work strengthened the hardness picture in the absence of DIMs. For a connected DIM-less graph, deciding whether it admits a non-trivial PED-set is NP-complete. Since every graph has the trivial PED-set $G$0, this implies that deciding whether a graph has a proper PED-set is NP-complete, and that deciding whether a DIM-less graph has at least two PED-sets is NP-complete (Grippo et al., 4 Sep 2025). This result isolates the genuinely PED-specific difficulty after the efficient-edge-domination case has been excluded.

4. Polynomial-time solvable classes

The positive side of the theory is driven by graph classes in which the coloring constraints collapse the search space.

For connected cubic claw-free graphs, every vertex lies in some triangle, and if every vertex of a connected graph lies in a triangle then the graph has no proper perfect edge dominating set. The proof uses black propagation: once a black vertex appears, the triangle constraints force adjacent vertices in triangles to be black, and connectivity propagates this to all vertices. As a consequence, on connected cubic claw-free graphs the weighted perfect edge domination problem reduces to comparing a minimum-weight DIM, if one exists, with the trivial set $G$1, yielding an $G$2-time algorithm (Lin et al., 2017).

For $G$3-free graphs, the weighted problem admits a robust linear-time algorithm: on arbitrary input, the algorithm either returns a minimum-weight PED-set or exhibits an induced $G$4 (Lin et al., 2017). The approach uses the Bacsó–Tuza theorem that every connected graph contains an induced $G$5, a dominating clique, or a dominating $G$6. A principal vertex can be found in linear time or an induced $G$7 can be detected; from a principal vertex one obtains either a dominating induced $G$8, a dominating clique, or an induced $G$9. The non-trivial PED case reduces to a finite list of admissible color patterns on a dominating $e$0; if there is a dominating clique $e$1 with $e$2, then the only PED-set is the trivial one (Lin et al., 2017).

For circular-arc graphs, the minimum weighted perfect edge domination problem (MWPED) is solvable in $e$3 time (Lin et al., 2015). The algorithm is based on a circular-arc model $e$4 and a case analysis driven by the overlap parameters

$e$5

The main cases are: two arcs covering the entire circle, three arcs covering the entire circle, Helly circular-arc models with a point covered by at least four arcs, overlap exactly three, and overlap exactly two with cycle-and-leaf structure. In the high-overlap cases, the problem is reduced to interval-graph MWPED by cutting arcs and using weight-preserving transformations; in the overlap-two case, the structure reduces to a controlled cycle with pendant arcs (Lin et al., 2015). The same paper also gives a linear-time algorithm for minimum weighted efficient edge domination on circular-arc graphs, using the equivalences

$e$6

together with the fact that $e$7 remains circular-arc (Lin et al., 2015).

For $e$8-free graphs, there is a cubic-time algorithm for finding a minimum-cardinality PED-set, and the same framework adapts to the weighted version and to counting all PED-sets and all DIMs without increasing the asymptotic complexity (Grippo et al., 4 Sep 2025). The structural engine is the characterization that a graph is $e$9-free iff every connected induced subgraph contains either a dominating induced $P\subseteq E(G)$00 or a dominating complete bipartite subgraph, and such a dominating subgraph can be found in $P\subseteq E(G)$01 time (Grippo et al., 4 Sep 2025). On a dominating induced $P\subseteq E(G)$02, only five valid partial colorings occur; on a dominating complete bipartite subgraph, the algorithm distinguishes the cases of no black vertices, exactly one black vertex, and at least two black vertices, then propagates forced colors through the remaining components (Grippo et al., 4 Sep 2025).

5. Counting and extremal theory

The counting problem asks for the number $P\subseteq E(G)$03 of PED-sets of $P\subseteq E(G)$04. This topic became algorithmically and extremally explicit in recent work (Grippo et al., 4 Jul 2026).

For paths, the numbers satisfy

$P\subseteq E(G)$05

with

$P\subseteq E(G)$06

Hence

$P\subseteq E(G)$07

(Grippo et al., 4 Jul 2026).

Among all trees on $P\subseteq E(G)$08 vertices, the path is extremal: $P\subseteq E(G)$09 Equality holds iff $P\subseteq E(G)$10 when $P\subseteq E(G)$11, iff $P\subseteq E(G)$12 when $P\subseteq E(G)$13, and iff $P\subseteq E(G)$14 otherwise (Grippo et al., 4 Jul 2026). Stars satisfy

$P\subseteq E(G)$15

and this equals $P\subseteq E(G)$16 only for $P\subseteq E(G)$17; for $P\subseteq E(G)$18, paths strictly beat stars (Grippo et al., 4 Jul 2026).

For forests, PED-counts multiply over connected components: $P\subseteq E(G)$19 The extremal forest problem is therefore a balancing problem over component sizes. The exact extremal families are determined for all small $P\subseteq E(G)$20, and for $P\subseteq E(G)$21 the unique extremal forest on $P\subseteq E(G)$22 vertices is again $P\subseteq E(G)$23 (Grippo et al., 4 Jul 2026).

For chordal graphs, the decisive structural theorem is that in an extremal graph, any connected component containing a triangle must be exactly that triangle. Since

$P\subseteq E(G)$24

an extremal chordal graph decomposes into isolated triangles plus an extremal forest remainder (Grippo et al., 4 Jul 2026). The resulting bound is

$P\subseteq E(G)$25

where

$P\subseteq E(G)$26

Equality is characterized exactly by disjoint unions of triangles together with the optimal residual forest from the tree classification (Grippo et al., 4 Jul 2026).

The same work gives linear-time counting algorithms for PED-sets and DIMs in generalized series-parallel graphs and chordal graphs (Grippo et al., 4 Jul 2026). For generalized series-parallel graphs $P\subseteq E(G)$27, the dynamic program uses states $P\subseteq E(G)$28, with $P\subseteq E(G)$29, encoding terminal colors and degree types. The base graph $P\subseteq E(G)$30 has only

$P\subseteq E(G)$31

all others zero. Explicit recurrences are given for series-1, series-2, and parallel composition, and since the state space is fixed, bottom-up processing of the parse tree yields $P\subseteq E(G)$32 time (Grippo et al., 4 Jul 2026). For chordal graphs, maximal cliques of size at least $P\subseteq E(G)$33 force all their vertices to be black in any valid coloring; after peeling them off, the remaining pieces have clique number at most $P\subseteq E(G)$34 and are generalized series-parallel, so the same dynamic program applies (Grippo et al., 4 Jul 2026).

The counting theory also motivates an open extremal direction: the conjecture that among all connected graphs on $P\subseteq E(G)$35 vertices, the cycle $P\subseteq E(G)$36 maximizes the number of PED-sets. Supporting evidence includes the recurrence

$P\subseteq E(G)$37

and the fact that $P\subseteq E(G)$38 has fewer PED-sets than $P\subseteq E(G)$39 for all $P\subseteq E(G)$40 (Grippo et al., 4 Jul 2026).

6. Relation to adjacent domination notions

Perfect edge domination sits in a broader family of edge-dominating and edge-covering concepts, but several nearby notions are distinct.

The most immediate comparison is with ordinary edge domination. An edge dominating set only requires that every edge outside the set share a vertex with some selected edge; perfect edge domination adds the exactly one requirement for all unselected edges (Lin et al., 2017). This exactness makes the $P\subseteq E(G)$41-coloring method possible and is the source of both the added hardness and the stronger structure theorems.

A different refinement is edge cut domination. In a connected graph, a set $P\subseteq E(G)$42 is an edge cut dominating set if it is an edge dominating set and the subgraph $P\subseteq E(G)$43 is disconnected; the minimum size is $P\subseteq E(G)$44 (Fenstermacher et al., 2016). This notion is not perfect edge domination. The relationship stated in the source is conceptual: in a perfect edge dominating set, every edge outside the set is adjacent to exactly one edge in the set, whereas edge cut domination requires domination plus disconnection after removing the set. In that sense, edge cut domination is weaker in the “exactly one” direction and stronger in the connectivity direction (Fenstermacher et al., 2016).

Another neighboring line is the theory of well-edge-dominated graphs, where all minimal edge dominating sets have the same cardinality (Berg et al., 2024). This is again not perfect edge domination, but it is structurally close because every matching is a minimal edge dominating set, so every well-edge-dominated graph is equimatchable (Berg et al., 2024). Recent classification results include a complete description of connected well-edge-dominated graphs containing exactly one triangle,

$P\subseteq E(G)$45

and the outerplanar classification

$P\subseteq E(G)$46

(Berg et al., 2024). These results bear on PED indirectly, chiefly through the interaction between minimal edge domination, matchings, and triangle structure.

Algorithmically, parameterized work on Edge Dominating Set is also adjacent rather than identical. A notable framework enumerates minimal vertex covers up to size $P\subseteq E(G)$47, tracks a partition $P\subseteq E(G)$48, and obtains an $P\subseteq E(G)$49-time algorithm together with a kernel satisfying

$P\subseteq E(G)$50

(Xiao et al., 2011). The source explicitly notes that this does not enforce the stronger perfect condition by itself, but that the endpoint-tracking and branching architecture is structurally close to PEDS-style enumeration (Xiao et al., 2011).

Finally, there is a vertex–edge analogue, ve-domination, in which a vertex dominates an edge if it is incident with the edge or adjacent to one of its endpoints. The class of well-ve-dominated graphs studies uniformity of minimal ve-dominating sets, not classical edge-selected PED-sets (Büyükçolak, 13 Dec 2025). Its strongest perfect-like statement is a tree characterization: a reduced tree is well-ve-dominated iff there exists an independent set

$P\subseteq E(G)$51

such that every edge of the tree is ve-dominated by exactly one vertex of $P\subseteq E(G)$52 (Büyükçolak, 13 Dec 2025). This suggests a broader theme: unique edge coverage can be formulated either by selecting edges, as in perfect edge domination, or by selecting vertices, as in ve-domination, but the two theories are distinct.

Perfect edge domination is therefore best understood as one member of a tightly connected family: stronger than ordinary edge domination, weaker than efficient edge domination, orthogonal to cut-based refinements, and structurally related to well-domination and matching-uniformity phenomena. The literature shows that this intermediate position is mathematically rich: it admits exact $P\subseteq E(G)$53-coloring characterizations, sharp hereditary complexity dichotomies, robust linear-time algorithms on some classes, cubic-time algorithms on others, and a developing extremal and counting theory (Lin et al., 2017, Grippo et al., 4 Sep 2025, Lin et al., 2015, Grippo et al., 4 Jul 2026).