1-restricted Simple 2-matchings

(Extended Abstract)

David Hartvigsen1

University of Notre DameNotre Dame, IN 46556-5646

Abstract

A simple 2-matching in a graph is a subgraph all of whose nodes have degree ≤ 2. A simple2-matching is called k-restricted if every connected component has > k edges. We consider theproblem of finding a k-restricted simple 2-matching with a maximum number of edges, which isa relaxation of the problem of finding a Hamilton cycle in a graph. Our main result is a newpolynomial time algorithm for finding a 1-restricted simple 2-matching with a maximum numberof edges. We also present a Tutte-type existence theorem and a Tutte-Berge-type formula for thisproblem.

Keywords: matching, Hamilton cycle

1 Introduction

All graphs considered in this paper are undirected with no parallel edges orloops. A simple 2-matching in a graph is a subgraph all of whose nodes havedegree ≤ 2 (for the sake of brevity, we henceforth drop the adjective simple).Hence the connected components of a 2-matching are paths and cycles. Awell-studied problem in the literature is to find a 2-matching in a graph witha maximum number of edges, a so-called maximum 2-matching. A polynomialtime algorithm, Tutte-type theorem and Tutte-Berge-type theorem are knownfor this problem, among many other things (see [18] for an excellent survey). Inthis paper we present analogous results for a restricted version of this problem,which we describe next.

1 Email: Hartvigsen.1@nd.edu

Electronic Notes in Discrete Mathematics 18 (2004) 145–149

1571-0653/$ – see front matter © 2004 Elsevier B.V. All rights reserved.

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doi:10.1016/j.endm.2004.06.023

A k-restricted 2-matching is a 2-matching such that each connected com-ponent has > k edges. The k-restricted 2-matching problem is to find a k-restricted 2-matching with a maximum number of edges, which we call a max-imum k-restricted 2-matching. Hence the 0-restricted 2-matching problem isto find a maximum 2-matching and the 1-restricted 2-matching problem is tofind a maximum 2-matching that contains no “isolated edges.” Note that thek-restricted problem is a relaxation of the problem of finding a Hamilton cyclein a graph due to the following observation: A graph G with n nodes has aHamilton cycle if and only if G has a k-restricted 2-matching with n edges,where k > �n/2�.

The main result in this paper is a new polynomial time algorithm for the 1-restricted 2-matching problem (an algorithm for the 2-piece-packing problemin [12] can be shown to also solve this problem; however, it is more complex; seebelow for a further discussion). An interesting feature of the algorithm is thatit starts with a maximum 2-matching and modifies it to obtain a maximum1-restricted 2-matching.

Before describing our other results, let us introduce some notation. Letek(G) denote the number of edges in a maximum k-restricted 2-matching fora graph G. Note that n ≥ e0(G) ≥ e1(G) ≥ e2(G) ≥ · · ·. We characterizethose graphs G for which e0(G) = e1(G) (an analogous result, characterizinggraphs for which n = e0(G), was obtained by Belck [1] and Gallai [9], and,in a more general form, by Tutte [20]; an analogous result for matchings wasproved by Tutte [19]). We also give a formula for e0(G)−e1(G) (an analogousresult for n − e0(G) can be proved using a construction of Tutte [21] and theTutte-Berge theorem [2] (see [18]); the Tutte-Berge theorem is an analogousresult for matchings). We also show that in every graph there always exists amaximum 2-matching that can be transformed into a maximum 1-restricted2-matching by removing its isolated edges.

We next discuss a closely related problem that has been studied in theliterature. A Ck-free 2-matching is a 2-matching that contains no cycles oflength ≤ k. An algorithm to find a maximum C3-free 2-matching appearsin [10] and an algorithm to find a maximum C4-free 2-matching in bipartitegraphs appears in [11]. See also [17]. This problem is a relaxation of theHamilton cycle problem, in the same way the k-restricted 2-matching prob-lem is. The edge-weighted version of this problem on complete graphs canbe shown to provide increasingly accurate approximations for the travellingsalesman problem as k increases (see [6]). The C5-free 2-matching problem isknown to be NP-hard (see [3] for a proof due to Papadimitiou). Related workcan be found in [3], [4], and [7].

Let us discuss one other related area of research. A k-piece is a connected

D. Hartvigsen / Electronic Notes in Discrete Mathematics 18 (2004) 145–149146

graph with maximum degree equal to k. A k-piece packing in a graph is asubgraph whose connected components are k-pieces. The node-max k-piecepacking problem is to find a k-piece packing in a graph that contains a max-imum number of nodes; the edge-max k-piece packing problem is to find a k-piece packing in a graph that contains a maximum number of edges. Observethat the node-max and edge-max 1-piece packing problems are equivalent tothe classical matching problem. However, for higher values of k, the node-maxand edge-max problems are different from one another. Finally, note that theedge-max 2-piece packing problem is identical to the 1-restricted 2-matchingproblem.

For k ≥ 2, only the node-max version of the k-piece packing problem hasbeen considered in the literature. The node-max 2-piece packing problem wasrecently considered by Kaneko in [15], where he presented a Tutte-type the-orem. This result was extended in [16] by Kano, Katona, and Kiraly, wherethe authors presented a Tutte-Berge-type theorem. These results were furtherextended by Hartvigsen, Hell, and Szabo in [12], where the authors presenteda polynomial time algorithm and Tutte-type and Tutte-Berge-type theoremsfor the node-max k-piece packing problem. Finally, a Gallai-Edmonds decom-position theorem for the general problem is presented in [14] by Janata, Loebl,and Szabo.

It’s not difficult to see how the algorithm in [12] can be modified to handlethe edge-max k-piece packing problem. (No change in the algorithm is neces-sary for k = 2. For k > 2, change the definition of tip value of a galaxy in [12]from the minimum number of nodes in a tip of the galaxy to the minimumnumber of edges in a tip of the galaxy. The proof of the modified algorithm’svalidity in both cases must also be changed.) However, the algorithm in [12]is complex and the modified version is no simpler. The key factor in thiscomplexity is the need to identify so called “critical” subgraphs (called “blos-soms” by Edmonds in [5] for k = 1, “suns” in [15] for k = 2, and “galaxies”in [12] in general). For the edge-max case when k = 2, however, it turns outthat the general algorithmic approach in [12] can be carried out in a differ-ent, simpler way to produce an algorithm that does not directly involve these“critical” subgraphs. This is the origin of the algorithm in this paper for the1-restricted 2-matching problem. We note that solutions produced by thissimplified algorithm need not be optimal solutions to the node-max 2-piecepacking problem. We use the algorithm to prove our Tutte- and Tutte-Berge-type theorems, which are new results.

There is a fairly large literature addressing problems like the ones men-tioned above: algorithms for finding (node or edge) maximum subgraphs in agraph whose connected components are in special classes (and the structural

D. Hartvigsen / Electronic Notes in Discrete Mathematics 18 (2004) 145–149 147

properties of these subgraphs). Much of this work has been recently surveyedin [13] by Hell (see also [18]).

The following section contains statements of our main theorems. Thestatement of the algorithm, the proof of its validity and complexity, and proofsof the main theorems do not appear in this abstract, although we brieflydiscuss the underlying approach.

2 The Theorems

In this brief section we state our main theorems. Their proofs make use of thestructures in the algorithm and do not appear in this abstract. Theorems 1 and2 are “Tutte-type” and “Tutte-Berge-type,” respectively; they characterize therelationship between e0(G) and e1(G), for any graph G.

For a graph G = (V,E) and T ⊂ V , we let G\T denote the graph obtainedfrom G by deleting the nodes in T (and all edges incident to these nodes). Aconnected component of G that contains exactly one edge is called an isolatededge of G. We let Edges(G) denotes the number of isolated edges in G. If Mis a 2-matching, then the 2-matching obtained by removing the isolated edgesfrom M is called the 1-restricted 2-matching associated with M .

Theorem 2.1 For any graph G = (V,E), e0(G) = e1(G) if and only if

2 ∗ |T | ≥ Edges(G\T ), for every T ⊂ V.

Theorem 2.2 For any graph G = (V,E),

e0(G) − e1(G) = maxT⊂V

{Edges(G\T ) − 2 ∗ |T |} .

Theorem 2.3 In every graph there exists a maximum 2-matching whose as-sociated 1-restricted 2-matching is maximum.

As mentioned in the introduction, Belck [1] characterized the graphs forwhich n = e0(G). Thus we can combine this result with Theorem 2.1 to obtaina characterization of the graphs for which n = e1(G). Also, we mentioned thata reduction due to Tutte [21] together with the Tutte-Berge formula [2] yieldsan expression for n − e0(G) (see [18]; this expression is typically written as amin-max formula). With this we can obtain a formula for n−e1(G). However,Belck’s condition and the expression for n− e0(G) are more complex than ourresults in Theorems 2.1 and 2.2 in that they involve two disjoint node setsinstead of our use of the one node set T .

Our algorithm for the maximum 1-restricted simple 2-matching problemhas the same general structure as Edmonds’ classical matching algorithm [5],without the need for shrinking. We define, for example, notions analogous to“alternating trees,” “augmenting paths,” and “augmentating the matching.”

D. Hartvigsen / Electronic Notes in Discrete Mathematics 18 (2004) 145–149148

The structure of the algorithm is used to prove the above theorems and tosimultaneously prove the validity of the algorithm.

References

[1] H.-B. Belck, Regulare Faktoren von Graphen, Journal fur die reine und angewandteMathematik 188 (1950) 228-252.

[2] C. Berge, Sur le couplate maximum d’un graphe, Comptes Rendus Hebdomadaires des Seancesde l’Academie Sciences [Paris] 247 (1958) 258-259.

[3] G. Cornuejols and W.R. Pulleyblank, A matching problem with side conditions, Discrete Math.29 (1980) 135-159.

[4] W.H. Cunningham and Y. Wang, Restricted 2-factor polytopes, Mathematical Programming87 (2000) 87-111.

[5] J. Edmonds, Paths, trees and flowers, Canad. J. of Math. 17 (1965) 449-467.

[6] M.L. Fisher, G.L. Nemhauser and L.A. Wolsey, An analysis of approximations for finding amaximum weight Hamiltonian circuit, Operations Res. 27, 4 (1979) 799-809.

[7] A. Frank, Restricted t-matchings in bipartite graphs, Discrete Appl. Math. 131, 2, (Sept. 2003)337-346.

[8] H.N. Gabow, An efficient reduction technique for degree-constrained subgraph and bidirectednetwork flow problems, in: Proceedings of the Fifteenth Annual ACM Symposium on Theoryof Computing (Boston, 1983), The Association for Computing Machinery, New York, 1983,448-456.

[9] T. Gallai, On factorisation of graphs, Acta Mathematica Academiae Scientiarum Hungaricae1 (1950) 133-153.

[10] D. Hartvigsen, Extensions of Matching Theory, Ph.D. Thesis, Department of Mathematics,Carnegie-Mellon U. (1984).

[11] D. Hartvigsen, Finding maximum square-free 2-matchings in bipartite graphs, manuscript(2000).

[12] D. Hartvigsen, P. Hell, J. Szabo, The k-piece packing problem, manuscript (2004).

[13] P. Hell, Packing in graphs, Electronic Notes in Discrete Mathematics, 5 (2000).

[14] M. Janata, M. Loebl, and J. Szabo, A Gallai-Edmonds type theorem for the k-piece packingproblem, manuscript (2004).

[15] A. Kaneko, A necessary and sufficient condition for the existence of a path factor everycomponent of which is a path of length at least two, J. Combin. Theory, Series B 88 (2003)195-218.

[16] M. Kano, G.Y. Katona, and Z. Kiraly, Packing paths of length at least two, Discrete Math.,to appear.

[17] Y. Nam, Matching Theory: Subgraphs with Degree Constraints and other Properties, Ph. D.Thesis, Department of Mathematics, University of British Columbia (1994).

[18] A. Schrijver. Combinatorial Optimization, Polyhedra and Efficiency. Springer 2003.

[19] W.T. Tutte, The factorization of linear graphs, J. London Math. Soc. 22 (1947) 107-111.

[20] W.T. Tutte, The factors of graphs, Canad. J. Math. 4 (1952) 314-328.

[21] W.T. Tutte, A short proof of the factor theorem for finite graphs, Canad. J. Math. 6 (1954)347-352.

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