An Improved Algorithm for Parameterized Edge Dominating Set Problem

. An edge dominating set of a graph G = ( V; E ) is a subset M ⊆ E of edges such that each edge in E \ M is incident to at least one edge in M . In this paper, we consider the parameterized edge dominating set problem which asks us to test whether a given graph has an edge dominating set with size bounded from above by an integer k or not, and we design an O ∗ (2 : 2351 k ) -time and polynomial-space algorithm. This is an improvement over the previous best time bound of O ∗ (2 : 3147 k ) . We also show that a related problem: the parameterized weighted edge dominating set problem can be solved in O ∗ (2 : 2351 k ) time and polynomial space.

EDS and PEDS are related to the vertex cover problem.A vertex cover of a graph is a set of vertices such that each edge of the graph is incident to at least one vertex in the set.The set of endpoints of all edges in any edge dominating set is a vertex cover.To find an edge dominating set of a graph, we may enumerate vertex covers of the graph and construct edge dominating sets from the vertex covers.Many previous algorithms are based on enumeration of vertex covers.We enumerate candidates of such edge dominating sets by branching on a vertex: fixing it as a vertex incident on at least one edge in an edge dominating set with a bounded size or not.In the O * (2.3147 k )-time algorithm to PEDS, Xiao et al. [9] observed that branching on vertices in a local structure called "2-path component" is the most inefficient among branchings on other local structures, and that reducing the number of branchings on 2-path components leads to an improvement over the time complexity.For this, they retained branching on 2-path components until no other structure remains, and effectively skipped subinstances that will not deliver edge dominating sets with a bounded size by systematically treating the set of 2-path components.In this paper, identifying new local structures, called "bi-claw," "leg-triangle" and "tri-claw components" and establishing a refined lower bound on the size of edge dominating sets, we design an O * (2.2351 k )time and polynomial-space algorithm.
Section 2 gives some terminologies and notations and introduces our branching operations of our algorithm.After Section 3 describes our algorithm that consists of three major stages, Section 4 analyzes the time complexity by deriving an upper bound on the number of all subinstances.Section 5 discusses a weighted variant of PEDS.Section 6 makes some concluding remarks.For space limitation, the proofs of lemmata are moved into Appendix A.
Lemma 1.Let k 1 , k 2 , . . ., k m be non-negative integers, where m ≥ 1.Then for any positive reals γ 1 , γ 2 , . .., γ m such that The set of vertices and edges in a graph H is denoted by V (H) and E(H), respectively.For a vertex v in a graph, let N (v) denote a set of neighbors of v and let N [v] denote a set of v and its neighbors (i.e., N [v] = {v} ∪ N (v)).A vertex of degree d is called a degree-d vertex.The degree of a vertex v in a graph H is denoted by d(v; H).For a set F of edges, we use V (F ) to denote a set of vertices incident on at least one edge in F , and we say that F covers a vertex set S ⊆ V if V (F ) ⊇ S. For a subset S ⊆ V of vertices, G[S] denote the subgraph of G induced by S. A cycle of length ℓ is called an ℓ-cycle, and is denoted by the sequence v 1 v 2 . . .v ℓ of vertices in it, where the cycle contains edges v 1 v 2 , . . ., v ℓ−2 v ℓ−1 and v ℓ v 1 .A connected component containing only one vertex is called trivial.We define five types of connected components as follows: a clique component, a connected component that is a complete subgraph; -a 2-path component, a connected component consisting of a degree-2 vertex u 1 and its two degree-1 neighbors u 0 , u 2 ∈ N (u 1 ), denoted by u 0 u 1 u 2 , as illustrated in Fig. 1(a); -a bi-claw component, a connected component consisting of two adjacent degree-3 vertices u 1 and v 1 and their four degree-1 neighbors u 0 , u 2 ∈ N (u 1 ) and v 0 , v 2 ∈ N (v 1 ), denoted by (u 0 u 1 u 2 )(v 0 v 1 v 2 ), as illustrated in Fig. 1(b); -a legged triangle component (or leg-triangle component), a connected component consisting of two adjacent degree-3 vertices u 1 and v 1 , their two degree-1 neighbors u 0 ∈ N (u 1 ) and v 0 ∈ N (v 1 ) and one common degree-2 neighbor w ∈ N (u 1 ) ∩ N (v 1 ), denoted by u 0 (u 1 wv 1 )v 0 , as illustrated in Fig. 1(c); and -a tri-claw component, a connected component consisting of three degree-3 vertices u 1 , v 1 and w 1 , their six degree-1 neighbors u 0 , u 2 ∈ N (u 1 ), v 0 , v 2 ∈ N (v 1 ) and w 0 , w 2 ∈ N (w 1 ) and their common degree-3 neighbor t ∈ N (u 1 ) ∩ N (v 1 ) ∩ N (w 1 ), denoted by t(u 0 u 1 u 2 )(v 0 v 1 v 2 )(w 0 w 1 w 2 ), as illustrated in Fig. 1(d).
The last four types of components, 2-path, bi-claw, leg-triangle and tri-claw components are called bad components collectively.We use two kinds of fundamental branching operations.One is to branch on an undecided vertex v ∈ U in (C, D): fix v as a new covered vertex in the first branch or as a new discarded vertex in the second branch.This is based on the fact that there is a (C, D)-eds M with v ∈ V (M ) or there is no such (C, D)-eds.Then we also fix all the vertices in N (v) as covered vertices in the second branch, since any edge e = vw incident to v needs to be incident to an edge dominating set at the vertex w.The other is to branch on a 4-cycle v 0 v 1 v 2 v 3 over undecided vertices: fix vertices v 0 and v 2 as new covered vertices or fix vertices v 1 and v 3 as new covered vertices.This is based on the fact that for any edge dominating set M , the set V (M ) is a vertex cover and one of {v 0 , v 2 } and {v 1 , v 3 } is contained in any vertex cover [8].Rooij and Bodlaender [7] found the following solvable case.

Lemma 2. [7] A minimum (C, D)-eds of an instance (C, D) such that G[U ] contains only clique components can be found in polynomial time.
We denote by U 1 the set of vertices of all clique components in G[U ], and let By Lemma 2, we only need to select vertices from U 2 to apply branching operations until all instances become leaf instances.
The next lower bound on the size of (C, D)-edses is immediate since for each clique component

Lemma 3. For any (C, D)-eds M in a graph G, it holds that
Based on this, we define the measure µ of an instance (C, D) to be We do not need to generate any instances (C, D) with µ(C, D) < 0 since they are not k-feasible.In this paper, we introduce the following new lower bound.

The Algorithm
Given a graph G and an integer k, our algorithm returns TRUE if it admits an edge dominationg set of size ≤ k or FALSE otherwise.The algorithm is designed to be a procedure that returns TRUE if a given instance (C, D) is k-feasible or FALSE otherwise, by branching on a vertex/4-cycle/bad component in (C, D) to generate new smaller instances (C (1) , D (1) ), . . ., (C (r) , D (r) ), to each of which the procedure is recursively applied.The procedure is initially given an instance (∅, ∅), and always returns FALSE whenever µ(C, D) < 0 holds.
Our algorithm takes three stages.The first stage keeps branching on vertices of degree ≥ 4, and retains the set B of all the produced bad components without branching on them.The second stage keeps branching on optimal vertices of degree ≤ 3, immediately branching on any newly produced bad component before it chooses the next optimal vertex to branch on.The third stage generates leaf instances by fixing all undecided vertices in the bad components in B, where we try to decrease the number of leaf instances to be generated based on some lower bound on the size of solutions of leaf instances.To derive the lower bounds in the third stage, we let C i store all vertices fixed to covered vertices during branching operations in the i-th stage.Formally EDSSTAGE1 is described as follows.When no vertex is left in U 2 \ V (B), we switch to the third stage.Formally EDSSTAGE2 is described as follows.

Algorithm EDSSTAGE1(C, D)
To obtain a leaf instance from the instance I, we need to fix all vertices in V (B).The number of all leaf instances that can be constructed from the instance , where r i is the number of subinstances generated by branching on a bad component H ∈ B i .
In the third stage, we avoid constructing of some "k-infeasible" leaf instances among all leaf instances.For a leaf instance , where C 3 denotes the set of undecided vertices in V (B) that are fixed to covered vertices in I ′ , we let w i,j be the number of bad components in B i to which the j-th branch is applied to generate I ′ , and call the vector w with these 16 entries w i,j the occurrence vector of I ′ .Note that ∑ i,j α i,j w i,j = |C 3 | holds, and that ∑ i,j β i,j w i,j is a lower bound on the size of (C 3 , D ′ )-eds by Lemma 4, since no edge in G joins two components in B. We derive two necessary conditions for a vector w to be the occurrence vector of a k-feasible leaf instance ( Observe that there is no edge between C 3 and C 2 in I ′ , since any vertex in ] during an execution of EDSSTAGE2.Hence ∑ i,j β i,j w i,j + ⌈|C 2 |/2⌉ is a lower bound on the size of a (C 3 ∪ C 2 , D ′ )-eds by Lemma 4, and another necessary condition is given by ( Note that the number ℓ(w) of leaf instances I ′ whose occurrence vectors are given by w is ) . ( For each instance , the third stage EDSSTAGE3 generates an occurrence vector w satisfying the conditions (1) and (2) and , and constructs all leaf instances the vector w, before it returns TRUE if one of the leaf instances is k-feasible or FALSE otherwise.Formally EDSSTAGE3 is described as follows.

The Analysis
For a given instance (G, k) of PEDS, let I i , i = 1, 2, 3 be the set of all instances constructed immediately after the i-th stage during the execution of EDSSTAGE1(∅, ∅), where I 3 is the set of all leaf instances, which correspond to the leaf nodes of the search tree of the execution.To analyze the time complexity of our algorithm, it suffices to derive an upper bound on |I 3 |.

Lemma 6. For any non-negative integer x 2 and an instance
From these, we obtain the next.Lemma 7.For any non-negative integers x 1 and x 2 , the number of instances

Note that the number of combinations
the number of possible occurrence vectors w satisfying the conditions (1) and (2) and ∑ j w i,j = y i , 1 ≤ i ≤ 4 is also bounded by a polynomial of n.To show that our algorithm runs in O * (2.2351 k ) time, it suffices to prove that the number of leaf instances generated from an instance w) denote the set of all such leaf instances.By Lemma 7 and (3), we see that In what follows, we derive an upper bound on O(1.380278 x1 •1.494541 x2 •ℓ(w)) under the constraints (1) and (2).For this, we merge some entries in w into ten numbers by which is bounded from above by an exponential function in Lemma 1 for any positive reals Then we have which is bounded by 1  11   , γ , γ , γ , γ , γ , γ , γ for any constants c 1 , c 2 and {c i,j } such that Conditions ( 1) and ( 2) are restated as As a linear combination of ( 6) and ( 7) with λ and (1 − λ), we get (5) for constants

A Related Problem: The Parameterized Weighted Edge Dominating Set Problem
We also consider a weighted variant of PEDS.The weighted edge dominating set problem (WEDS) is, given a graph G = (V, E) with an edge weight function ω : E → R ≥0 , to find an edge dominating set M of minimum total weight ω(M ) = ∑ e∈M ω(e).The parameterized weighted edge dominating set problem (PWEDS) is, given a graph G = (V, E) with an edge weight function ω : E → R ≥1 and a positive real k, to test whether there is an edge dominating set M such that ω(M ) ≤ k.We show that a modification of our algorithm for PEDS can solve PWEDS in the same time and space complexities as our algorithm does PEDS.
For PWEDS we use the same terminologies and notations as for PEDS; for example, an instance of PWEDS is also denoted by (C, D).Rooij and Bodlaender [7] found the following solvable case for a weighted variant of EDS.

Lemma 8. [7] A minimum (C, D)-eds of an instance (C, D) of WEDS such that G[U ] contains only clique components of size ≤ 3 can be found in polynomial time.
Based on this lemma, for PWEDS we modify U 1 to be the set of vertices of clique components of size ≤ 3 in G[U ].We call our algorithm to which this modification is applied a modified algorithm.This modification brings the following corollary.⊓ ⊔

Conclusion
In this paper, we have presented an O * (2.2351 k )-time and polynomial-space algorithm to PEDS.The algorithm retains bad components produced at the first stage for branching on vertices of degree ≥ 4, and branching on the remaining undecided vertices not in clique components by choosing 4-cycles/vertices to branch on carefully.Based on our new lower bound on the size of (C, D)-edses, we derived an upper bound on the number of leaf instances generated in the third stage.We have also shown that a modification of our algorithm can solve PWEDS in the same time and space complexities as PEDS.For a possible achievement of further improved algorithms, it is still left to modify the first stage of our algorithm to branch on vertices of degree ≤ 4 in the second stage and to identify several new components as bad components.
Appendix A Lemma 1.Let k 1 , k 2 , . . ., k m be non-negative integers, where m ≥ 1.Then for any positive reals γ 1 , γ 2 , . .., γ m such that Proof.We proceed by an induction on ∑ m i=1 k i to prove the lemma.I.The lemma holds when ∑ m i=1 k i = 0, since the both sides of the inequality in the lemma become 1.II.Assume that the lemma holds for any instance ∑ m i=1 k i ≤ K for some integer K ≥ 0. We show that the lemma holds for any instance {k 1 , k 2 , . . ., k m } with ∑ m i=1 k i = K + 1.If k j = 0 for some j, where m ≥ 2 by ∑ m i=1 k i = K + 1 > 0, then it suffices to show that the lemma holds for the instance {k 1 , k 2 , . . ., k m } \ {k j }, since γ kj j = 1 for any choice of {γ 1 , γ 2 , . . ., γ m }.Hence we assume without loss of generality that k i ≥ 1 for all i = 1, 2, . . ., m.Let γ 1 , γ 2 , . . ., γ m satisfy ∑ m i=1 1/γ i ≤ 1.Using Pascal's rule and the inductive hypothesis, we obtain the following inequality: This proves that the lemma also holds for any instance In what follows, we prove Lemmata 5 and 6.Let T (µ) be the maximum number of leaf instances that can be generated from an instance I with measure µ.

Lemma 5. For any non-negative integer x 1 , the number of instances
Proof.At the first stage, the algorithm branches on a vertex v of degree ≥ 4 in G[U 2 ].When the algorithm branches on v by fixing it as a covered vertex or a discarded vertex, {v} (resp., N (v)) is added to the set C, and the measure µ decreases by 1 (resp., |N (v)| ≥ 4).Hence we have the following recurrence: which solves to T (µ) = O(1.380278µ ).This proves the lemma.

Lemma 6.
For any non-negative integer x 2 and an instance I = (C 1 , ∅, B, D) ∈ I 1 , the number of instances We use U ′ 2 to denote U 2 \ V (B).To prove Lemma 6, we derive recurrences for branchings executed by Algorithm EDSSTAGE2.We first show recurrences for branching on bad components only.

Lemma 9. Assume that Algorithm EDSSTAGE2 branches on a bad component H in G[U
. If H is a 2-path component, then the algorithm branches on H with the following recurrence:

which solves to T (µ) = O(1.6181 µ ). If H is a bi-claw or leg-triangle component, then the algorithm branches on H with the following recurrence:
which solves to T (µ) = O(1.5214µ ).If H is a tri-claw component, then the algorithm branches on H with the following recurrence: Proof.In the i-th branch of each bad component H, all vertices in C (i) (H) are fixed as covered vertices and thereby the measure decreases by |C (i) (H)|.Therefore we have the above recurrences.

⊓ ⊔
Observe that Algorithm EDSSTAGE2 branches on a bad component with the recurrence shown in Lemma 9, which is not good enough to establish Lemma 6.In our analysis, we combine a branching on a bad component together with the branching on the optimal vertex v (or the admissible 4-cycle on it) that produces the bad component, which yields a recurrence better than those in Lemma 9.In the case where the branching on v and the all bad components produced by any of the branchings to v yields a recurrence even not good enough to establish Lemma 6, we further combine it with a possible branching on a vertex of condition (c-1), (c-2) or (c-3)(iv) produced by the branching to v.In what follows, for each i = 1, 2, . . ., 6 in this order, we analyze the branching of an optimal vertex v satisfying condition (c-i) to derive such a recurrence.

Lemma 10. Algorithm EDSSTAGE2 branches on a vertex v satisfying condition (c-1) in G[U ′
2 ] together with possible branchings on the resulting new bad components with the following recurrence: which solves to T (µ) = O(1.494541µ ).
Proof.Since v is a vertex satisfying condition (c-1), v is a degree-3 . Neither of the first and second branches produces a new bad component.Therefore the algorithm branches on v with the following recurrence: which solves to T (µ) = O(1.4656µ ) and is better than the recurrence (8).
⊓ ⊔ Lemma 11.Algorithm EDSSTAGE2 branches on an optimal vertex satisfying condition together with possible branchings on the resulting new bad components with a recurrence not worse than (8).
Proof.Since v is an optimal vertex satisfying condition (c-2), v is a degree-2 (x, y)-vertex with x+y ≤ 1 and . We distinguish two cases: Case 1. x + y = 0; and Case 2. x + y = 1.Case 1. x = y = 0: In any of the first and second branches, no bad component is newly produced.Therefore the algorithm branches on v with the following recurrence: which solves to T (µ) = O(1.4143µ ).

Case 2. x + y = 1:
In one of the first and second branches, exactly one bad component H is newly produced, and then the algorithm branches on it; and in the other branch, no bad component is newly produced.In the following, we derive recurrences for branching on v together with branching on H.When H is a 2-path component, we have the following recurrence: which solves to T (µ) = O(1.4656µ ).When H is a bi-claw or leg-triangle component, we have the following recurrence: which solves to T (µ) = O(1.4560µ ).When H is a tri-claw component, we have the following recurrence: which solves to Since all the recurrences obtained in Cases 1 and 2 are better than the recurrence ( 8), the lemma holds.
⊓ ⊔ Lemma 12. Algorithm EDSSTAGE2 branches on an optimal vertex satisfying condition together with possible branchings on the resulting new bad components with a recurrence not worse than (8).
Proof.Since v is an optimal vertex satisfying condition (c-3), v is in one of the following four cases: which solves to T (µ) = O(1.4143µ ); and which solves to T (µ) = O(1.3803µ ), and at most one bad component H is newly produced in one of the first and second branches.We consider three subcases (a)-(c).

Case (a). The algorithm branches on v (or the admissible 4-cycle on it) in G[U ′
2 ] with the recurrence (12) and exactly one bad component H is produced in one of the first and second branches: When H is a 2-path component, we have the recurrence (9).When H is a bi-claw or leg-triangle component, we have the recurrence (10).When H is a tri-claw component, we have the recurrence (11).
Case (b).The algorithm branches on v in G[U ′ 2 ] with the recurrence ( 13) and exactly one bad component H is produced in the first branch: When H is a 2-path component, we have the following recurrence: When H is a bi-claw or leg-triangle component, we have the following recurrence: which solves to T (µ) = O(1.4527µ ).When H is a tri-claw component, we have the following recurrence: Case (c).The algorithm branches on v in G[U ′ 2 ] with the recurrence ( 13) and exactly one bad component H is produced in the second branch: When H is a 2-path component, we have the following recurrence: When H is a bi-claw or leg-triangle component, we have the following recurrence: When H is a tri-claw component, we have the following recurrence: Case (iii): When x = y = 0; i.e., neither of the first and second branches produces a new bad component, the algorithm branches on v with the following recurrence: Consider the case where x + y = 1; i.e., one of the first and second branches produces exactly one new bad component H other than a 2-path component whereas the other branch produces no new bad component.The algorithm branches on v together with branching on H with one of the following four recurrences.When x = 1, y = 0 and H is a bi-claw or leg-triangle component, we have which solves to T (µ) = O(1.494541µ ).When x = 1, y = 0 and H is a tri-claw component, we have which solves to T (µ) = O(1.4914µ ).When x = 0, y = 1 and H is a bi-claw or leg-triangle component, we have which solves to T (µ) = O(1.4841µ ).When x = 0, y = 1 and H is a tri-claw component, we have which solves to T (µ) = O(1.4842µ ).

Case (iv):
In the first branch, no bad component and a degree-3 (0, 0)-vertex u are newly produced, and then the algorithm branches on u, since u satisfies condition (c-1) after fixing v as a covered vertex.
In the second branch, exactly one 2-path component is newly produced.Therefore the algorithm branches on v together with branching on u and the 2-path component with the following recurrence: which solves to T (µ) = O(1.4865µ ).Since all the recurrences obtained in Cases (i)-(iv) are not worse than the recurrence (8), the lemma holds.

⊓ ⊔
We say that an instance has no vertices of degree ≥ 4, no vertices satisfying any of conditions (c-1) to (c-i) and no bad components.Lemma 13.Let (C, D) be an instance reduced up to (c-3).
Proof.(i) Now the degree of every vertex in by the assumption on (C, D).We first prove the next claim.
, where 1 ≤ k ≤ 3. We distinguish three cases k = 1, Case 1. k = 1: Without loss of generality there are four cases: (a) H is a bi-claw component and u 0 is adjacent to v, where these four cases are illustrated in Fig. 2. If v is a degree-2 vertex and has a degree-1 neighbor in Case (a), (b) or (d), then u 0 is a vertex with q u0 = 1 in G[U ′ 2 ], which satisfies (c-2).Assume that v is not such a vertex.We show that the degree-3 vertex v 1 ∈ V (H) furthest from v satisfies (c-1) or (c-3). Cases Case (d): The degree-3 vertex v 1 satisfies both of the following two conditions: removing v 1 from G[U ′ 2 ] produces a degree-3 (0, 0)-vertex w 1 ; and removing produces exactly one 2path component.Thus v 1 satisfies (c-3)(iv).
Case 2. k = 2: Without loss of generality there are six cases: (a) H is a bi-claw component where these six cases are illustrated in Fig. 3.If v has a degree-1 neighbor in G[U ′ 2 ], then v is a degree-3 (1, 0)-vertex such that removing v from G[U ′ 2 ] produces exactly one bad component, i.e., H, which is not a 2-path component.Hence v satisfies (c-3)(iii).Assume that v is not such a vertex.We show that the degree-3 vertex v 1 ∈ V (H) furthest from v satisfies (c-1) or (c-3).
Cases (a), (b), (c) and (d): The degree-3 vertex v 1 satisfies both of the following two conditions: removing v 1 from G[U ′ 2 ] produces no bad component; and removing produces at most one bad component other than a 2-path component.Therefore v 1 satisfies (c-1) or (c-3)(iii).
This prove the claim.

⊓ ⊔
Next we prove that the set of new bad components in G[U ′ 2 \ {v}] is a set of three 2-path components.Let P 1 , P 2 , . . ., P bv be the new bad components produced in G[U ′ 2 \ {v}], all of which are 2-path components.To prove the property (i) of the lemma, we assume that b v ∈ {1, 2}, and prove that some neighbor of v satisfies one of conditions (c-1) to (c-3) in G[U ′ 2 ].Without loss of generality for the 2-path component P 1 = v 0 v 1 v 2 , there are the following five cases: (a) and (e) N (v) ⊆ V (P 1 ), as illustrated in Fig. 4. For Case (d) or (e), there is an admissible 4-cycle vv , implying that v satisfies condition (c-3)(i).Assume that neither of Case (d) and (e) holds for P 2 if any.
Next consider Case (a).We see that , removing v 0 produces new 2-path components.Hence v 0 is a degree-2 (x, y)-vertex with x + y ≤ 1 and , satisfying condition (c-2).Assume that Case (a) does not hold for P 2 if any.
Finally consider Case (b) or (c).Let H denote the component containing and denote P 2 by w 0 w 1 w 2 , where w 1 ∈ N (u) and P 2 satisfies configuration (b) or (c).We show that v 1 satisfies condition (c-3 Hence v 1 is a vertex satisfying condition (c-3)(iv), as required.
(ii) Let u be a degree-2 vertex with produces a set of three 2-path components by (i), which also indicates that v is of degree 3 in G[U ′ 2 ].Conversely let v be a degree-3 vertex removal of which produces exactly three since removing u 0 produces the clique component consisting of {u 1 , u 2 }. ⊓ ⊔ Lemma 14. Algorithm EDSSTAGE2 branches on an optimal vertex v satisfying condition (c-4) in G[U ′ 2 ] together with possible branchings on the resulting new bad components with a recurrence not worse than (8).
Proof.Since v is an optimal vertex satisfying condition (c-4), v is a degree-2 vertex with produces exactly two components: the component H ′ containing u and the clique component Q of size 2. Now Lemma 13 holds for (C, D), and v has a degree-3 neighbor u removal of which produces exactly three 2-path components P 1 , P 2 and P 3 .We see that the component H containing v is a graph consisting of P 1 , P 2 and P 3 and the degree-3 vertex u adjacent to all these 2-path components, one of which say P 3 is given by vv ′ v ′′ for {v ′ , v ′′ } = V (Q).Let w i , i = 1, 2, be the neighbor of u in P i .In what follows, we show that the algorithm continues to branch on one of w 1 and w 2 , say w after fixing v as a covered vertex, and branches on the other of them after fixing w as a covered vertex, and then derive recurrences for branching on v together with branchings on w, w ′ and all newly produced bad components.Without loss of generality, Case (a).d(w 1 ; H) = d(w 2 ; H) = 3: From the structure of H, we see that w 1 is a degree-3 (0, 1)- After v is fixed as a covered vertex, the algorithm branches on one of them, say w and continues to branch on the other of them after fixing w as a covered vertex with the recurrence (14).Therefore we have the following recurrence: Case (c).d(w 1 ; H) = d(w 2 ; H) = 2: From the structure of H, we see that w 1 is a degree-2 (0, 1)vertex with q w1 = 1 in G[U ′ 2 \ {v}] such that removing w 1 from G[U ′ 2 \ {v}] changes w 2 to a degree-2 (0, 0)-vertex with q w2 = 1; and removing N [ ] contains a 6-cycle such that either (a) vu 0 u 1 u 2 u 3 u 4 consisting of v and five degree-2 vertices u i , i = 0, 1, 2, 3, 4; or (b) vv ′ u 0 u 1 u 2 u 3 consisting of v, another degree-3 (0, 1)-vertex v ′ , and four degree-2 vertices u i , i = 0, 1, 2, 3. (iv) For any degree-3 (0, 2)-vertex v in G[U ′ 2 ] of (C, D), the component H containing v in G[U ′ 2 ] consists of either (c) two 6-cycles vv ′ u 0 u 1 u 2 u 3 and vv ′ v 0 v 1 v 2 v 3 that share an edge vv ′ between v and another degree-3 (0, 2)-vertex v ′ and pass through degree-2 vertices u i and v i , i = 0, 1, 2, 3; or (d) a 4-cycle vv 0 v 1 v 2 of v and three other degree-3 (0, 2)-vertices v i , i = 0, 1, 2 and two paths vu 0 u 1 u 2 v 1 and v 0 w 0 w 1 w 2 v 2 joining two vertices in the 4-cycle and passing through degree-2 vertices u i and w i , i = 0, 1, 2.  Proof.Now the degree of every vertex in U ′ 2 is at most 3 in G[U ′ 2 ] by the assumption on (C, D). (i) Lemma 13 holds due to the assumption, and there is no degree-2 vertex u with q u = 1 in G[U ′ 2 ].Therefore for any vertex v in G[U ′ 2 ], removing v from G[U ′ 2 ] produces no bad component.

The four types
Lemma 18. Algorithm EDSSTAGE2 branches on an optimal vertex v satisfying condition (c-6) in G[U ′ 2 ] together with possible branchings on the resulting new bad components with a recurrence not worse than (8).
Proof.Since v is an optimal vertex satisfying condition (c-6), v is a degree-2 vertex in G[U ′ 2 ].Let H be the component containing v in G[U ′ 2 ].In the following, we show that H is a cycle component of length ≥ 4.
Since there is no vertex that satisfies condition (c-5), there are only vertices of degree ≤ 2 in G[U ′ 2 ].Furthermore there is no vertex of degree ≤ 1 in G[U ′ 2 ], since G[U ′ 2 ] has no clique component, no 2-path component and no degree-2 vertex u with q u ≥ 1, which satisfies condition (c-2).Therefore there are only degree-2 vertices in G[U ′ 2 ], indicating that the component containing v in G[U ′ 2 ] is a cycle component of length ≥ 4.
Algorithm EDSSTAGE2 branches on vertices of H until G[U ′ 2 ] has no more vertices of H, with a recurrence not worse than (8), by Lemma 17.
⊓ ⊔ Now we are ready to complete the proof of Lemma 6. Lemmata 10, 11, 12, 14, 16 and 18 guarantee that Algorithm EDSSTAGE2 branches on an admissible 4-cycle or an optimal vertex in G[U ′ 2 ] together with possible branchings on the resulting new bad components with a recurrence not worse than (8).⊓ ⊔

Input:
A graph G = (V, E) with an integer k, and subsets C and D of V (initially, C = D = ∅).Output: TRUE if (C, D) is k-feasible or FALSE otherwise.1: if µ(C, D) < 0 then 2: return FALSE 3: else if there is a vertex v of degree ≥ 4 in G[U2] then 4: return EDSSTAGE1(C ∪ {v}, D) ∨ EDSSTAGE1(C ∪ N (v), D ∪ {v}) 5: else 6: C1 := C; C2 := ∅; 7: Let B store all bad components in G[U2]; 8: return EDSSTAGE2(C1, C2, B, D) 9: end if For a given instance (G, k) of PEDS, let I 1 denote the set of all instances constructed immediately after the first stage.Let V (B) denote the set of vertices in the bad components in B. Given an instance (C 1 , C 2 , B, D) ∈ I 1 , the second stage EDSSTAGE2 fixes all vertices in U 2 \ V (B) to covered/discarded vertices by repeatedly branching on optimal vertices or any newly produced bad component in G[U 2 \ V (B)] if it exists.During the second stage, the sets C 1 and B obtained in the first stage never change.
the set of all instances constructed immediately after the second stage.Consider an instance I = (C 1 , C 2 , B, D) ∈ I 2 , where the graph G[U 2 ] consists of the bad components in B retained at the first stage.Let B 1 (resp., B 2 , B 3 and B 4 ) be the sets of 2-path (resp., bi-claw, leg-triangle and triclaw) components in B, and

Corollary 1 .
The modified algorithm can solve the parameterized weighted edge dominating set problem in O * (2.2351 k ) time and polynomial space.Proof.We first show the correctness.If an edge dominating set M of G is k-feasible, i.e., ω(M ) ≤ k, then it holds that |V (M )| ≤ 2k and |M | ≤ k since ω(e) ≥ 1 for any edge e ∈ E. This ensures the correctness of the measure µ(C, D) and the conditions (1) and (2) for an instance (C, D) of the weighted variant.Therefore we can solve PWEDS by the same branching method as PEDS.Second we show the time complexity is the same as PEDS.Only difference between our algorithm for PEDS and one for PWEDS is treatment of clique components of size ≥ 4. In what follows, we describe the treatment by the modified algorithm and it guarantees that the time complexity is O * (2.2351 k ).For a clique component H of size ≥ 5 of an instance (C, D), the degree of a vertex of H in G[U 2 ] is |V (H)| − 1 ≥ 4, on which therefore the modified algorithm branches in the first stage.For a clique component H of size 4 of an instance (C, D), a vertex of H satisfies condition (c-2), on which therefore the algorithm branches in the second stage.
d and removing each of v and N[v] produces no new 2-path component; and (iv) v is a degree-3 (0, 1)-vertex such that removing N [v] produces exactly one new 2-path component, and G[U 2 \ {v}] contains at least one degree-3 (0, 0)-vertex.We distinguish three cases: Case (i) or (ii); Case (iii); and Case (iv).Case (i) or (ii): When the algorithm branches on v (or the admissible 4-cycle on it) in G[U ′ 2 ], we have one of the following two recurrences: (a), (b) and (c): The degree-3 vertex v 1 satisfies both of the following two conditions: removing v 1 from G[U ′ 2 ] produces no bad component; and removing N [v 1 ] from G[U ′ 2 ] produces at most one bad component other than a 2-path component.Therefore v 1 satisfies (c-1) or (c-3)(iii).

Fig. 3 .
Fig. 3. Components containing v such that a bi-claw, leg-triangle or tri-claw component H is produced by removing v and k= |N (v) ∩ V (H)| = 2 in G[U ′ 2 ] and there is only one bad component other than a 2-path component in G[U ′ 2 \ {v}].

Fig. 4 .
Fig. 4. Components containing v such that a 2-path component v0v1v2 is produced by removing v

Fig. 6 .
Fig. 6.Components containing a degree-3 vertex v under the assumption in Lemma 15