Parameterized Algorithms for Finding SmallIndependent Dominating Sets in Planar

Graphs

Faisal N. Abu Khzam 1,2

Division of Computer Science and Mathematics

Lebanese American University

Beirut, Lebanon

Henning Fernau 3

FB 4—Abteilung Informatik

Universitat Trier

54286 Trier, Germany

1 Introduction

Fixed-parameter tractability emerged as one of the most promising tools forcoping with computational intractability. The theory is concerned mainly withthe classification of computational complexity of parameterized NP problems,but has led implicitly to the development of many algorithmic techniques.One of the most popular methods is the bounded search tree technique [3].In brief, this technique is nothing but recursive backtracking where the search

1 This research has been supported in part by the Lebanese American University undergrant URC-2004-c632 Email: faisal.abukhzam@lau.edu.lb3 Email: henningfernau@yahoo.de

Electronic Notes in Discrete Mathematics 25 (2006) 1–6

1571-0653/$ – see front matter © 2006 Elsevier B.V. All rights reserved.

www.elsevier.com/locate/endm

doi:10.1016/j.endm.2006.06.049

for a solution is guided by a tree whose height is bounded by a function ofthe input parameter. So far, it seems the most usable input parameter is thesize of the sought solution. All such algorithms involve a case analysis that isused in making decisions at search tree nodes, and leads to determining thebranches of the tree that emerge from each node.

We are considering the following problem, termed independent dominating

set on planar graphs:Given: a planar graph G = (V, E) and a parameter k ≥ 0.Question: Is there an independent dominating set D ⊆ V with |D| ≤ k?

It is well-known that this question is equivalent to finding a maximal inde-pendent set of size at most k, i.e., to minimum maximal independent set. Thereis an obvious parameterized branching algorithm for this problem, based onthe observation that, for each v ∈ V , one vertex from the closed neighbor-hood N [v] must go into the dominating set. Euler’s formula guarantees thatthere is always a vertex of degree at most five in a graph, so that the sketchedalgorithm runs in time O∗(6k).

We present a search tree algorithm that is based on a deep insight intothe local structure of planar graphs (see Theorem 1). Instead of applyingthat theorem directly (which would only give a modest improvement overthe run-time estimate of a naive exact algorithm), we rather infer a corollarythat further settles the local structure of the “critical graphs” (those havingminimum degree of four or five).

2 A Structural Theorem

Our work in this paper is motivated and based upon the following structuraltheorem from [2].

Theorem 2.1 Every connected plane graph with at least two vertices has

(i) two vertices with degree sum at most 5, or

(ii) two vertices of distance at most two and with degree sum at most 7, or

(iii) a triangular face with two incident vertices with degree sum at most 9, or

(iv) two triangular faces neighbored via an edge {u, v} where the sum of thedegrees of u and v is at most 11.

To simplify our subsequent arguments, we use the following definitions.

Definition 2.2 A pair of vertices (u, v) is said to be a five-six pair if it satisfiesthe following:

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• u and v are adjacent,

• deg(u) = 5 and deg(v) = 6, and

• the edge {u, v} is common to two triangular faces.

Definition 2.3 The five-six degree of a vertex t is the number of five-six pairs(u, v) such that t is a common neighbor of u and v.

Definition 2.4 A vertex of degree smaller than six is said to be an isolated-low-degree vertex if all its neighbors are of degree seven or more.

Theorem 2.1 can be used to study the structure of the neighborhoods oflow degree vertices. A closer look at these neighborhoods yields the followingcorollary:

Corollary 2.5 Assume that G = (V, E) is a connected plane graph with atleast two vertices. If G has no isolated low degree vertices, then one of thefollowing cases occurs:

(i) There is one vertex of degree at most 3.

(ii) There are two adjacent vertices u, v where u has degree four and v hasdegree five such that edge {u, v} is common to (or borders) one triangularface.

(iii) There are two adjacent vertices u, v each of degree five, such that edge{u, v} is common to two triangular faces.

(iv) There is a five-six pair of vertices, (u, v) such that one of the followingconditions applies:(a) ∃t ∈ N(u) \ N(v) : deg(t) = 5.(b) ∃t ∈ N(v) \ N(u) : deg(t) = 5.(c) ∃t ∈ N(v) ∩ N(u) : deg(t) = 6.(d) ∃t ∈ N(v) ∩ N(u) such that the five-six degree of t is in {2, 3, 4, 5}.

Proof. If one of the first three cases of Theorem 2.1 applies, the first andsecond case of this corollary are covered. If not, the fourth case of Theorem 2.1applies. One possibility for this is the third case of this corollary. Otherwise,assume that the previous cases do not apply anywhere in the graph.

If all pairs {(u, v), (u′, v′)}, of five-six pairs are independent in the sensethat (N(u)∪N(v))∩ (N(u′)∪N(v′)) = ∅, then contracting all these edges uv,u′v′, . . . creates a planar graph. There must be a vertex of degree five in thisnew planar graph. Hence, for some vertex t from say (N(u) \N(v))∪ (N(v) \N(u))), deg(t) = 5. This covers cases 4(a) and 4(b). The same contractionargument shows that there could be also a vertex of degree six in N(u)∩N(v),

F.N. Abu Khzam, H. Fernau / Electronic Notes in Discrete Mathematics 25 (2006) 1–6 3

covering case 4(c).

If some five-six pairs are dependent on one another somehow, this meansthat (N(u)∪N(v))∩ (N(u′) ∪N(v′)) = ∅, and assuming cases 4(a), 4(b) and4(c) don’t apply, then the degree of some common neighbor, t, of a five-sixpair (u, v) must drop down to five or less (after the edge contraction opera-tions). This is possible only if t is a common neighbor of another five-six pair.Moreover, the five-six degree of t is a lower bound on its new degree (after thecontraction). This completes the proof.

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3 A rough sketch of the algorithm and its analysis

A trivial branching scenario would be to take a vertex or any of its neighborsinto the target independent set (henceforth I). If this is applied (blindly) onlow degree vertices, we get a run time of O∗(6k). Hence, we set it as ourobjective to get a better run time. To do this, we adopt a branching strategythat is based on taking vertices into I according to the following simple rule:

If u and v are adjacent (low-degree) vertices, then we have two choices:

• Take u or v or one of their common neighbors (if any) into I.

• Take one vertex from N(u)\N(v) and one vertex from N(v)\N(u) into I.

The idea of branching at small-degree vertices allows for a refined analysis.This is shown exemplarily by considering branching at vertices of degree atmost four:

(i) If deg(v) ≤ 3 for some vertex, we can take the trivial T (k) ≤ 4T (k − 1)-branching, i.e., T (k) ≤ 4k. Here, T (k) is the number of leaves in thesearch tree (each branching scenario has a corresponding upper boundon T (k)).

(ii) If deg(v) = 4 for some vertex, we can take the trivial T (k) = 5T (k − 1)-branching, i.e., T (k) ≤ 5k. However, due to theorem 2.1, the absence ofvertices of degree smaller than 4 dictates that v must have a neighbor,w, of degree 5,6 or 7 that shares at least one triangular face with v. Letus assume, for now, that the graph has no isolated low-degree vertices,then our corollary asserts that w is of degree 5. We have the followingsubcases:(a) If v has two common neighbors with w, then our branching strategy

gives 4T (k − 1) + 2T (k − 2). So T (k) ≤ 4.45k.(b) If v has one common neighbor with w, then we get 3T (k−1)+6T (k−

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2), and the corresponding T (k) is bounded above by 4.3723k.(c) If N(v) ⊂ N(w), then it can be shown that the graph has a separator

consisting of a cycle of length three that contains v and w. Thisallows us to have a better branching scenario, as we discuss below.

All in all, this case gives a branching time in O(4.45k).

The analysis of the degree-5 vertices is based on another algorithmic ideathat might be interesting in other contexts, as well.

The second main idea of the algorithm is to branch on small cycle separa-tors, i.e., cycles of length at most four whose removal splits the graph in twonon-trivial components. On cycles of length three, for example, we branch asfollows: either incorporate one vertex of the cycle in the independent domi-nating set we implicitly construct on each path of the search tree, or do notincorporate any of the cycle vertices. Since both components of the remaininggraph are non-trivial, processing the smaller one will reduce the budget thatremains for the bigger one, so that asymptotically we have a better run timethan in the case of branching on vertices.

We omit details of the further analysis as contained in the full paper andonly summarize our results as follows.

Theorem 3.1 minimum maximal independent set on planar graphs can besolved in time O∗(4.97k).

4 Conclusions

This paper is the second one in which we showed how a non-trivial corollaryof Theorem 2.1 can be used to considerably improve on known (or easy-to-derive) parameterized algorithms on planar graphs; the first example beingface cover, as published in [1].

It would be interesting to see improvements on other problems by makinguse of deep structural insights from graph theory (or other areas). Conversely,our results may stir the interest in developing such insights, having possiblealgorithmic applications in the back of one’s mind.

References

[1] F. N. Abu-Khzam, H. Fernau and M. A. Langston. Asymptotically fasteralgorithms for the parameterized face cover problem. In H. Broersma, M.Johnson and S. Szeider (editors): Algorithms and Complexity in Durham ACiD

F.N. Abu Khzam, H. Fernau / Electronic Notes in Discrete Mathematics 25 (2006) 1–6 5

2005, vol. 4 of Texts in Algorithmics, pp. 43–58. London: King’s CollegePublications, 2005.

[2] V. A. Aksionov, O. V. Borodin, L. S. Mel’nikov, G. Sabidussi, M. Stiebitz, andB. Toft. Deeply asymmetric planar graphs. J. Comb. Theory Ser. B, 95:68–78,2005.

[3] R. G. Downey and M. R. Fellows. Parameterized Complexity. Springer, 1999.

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