A Note on the Matching Polytope of a Graph

A Note on the Matching Polytope of a Graph

N.M.M. ABREU¹, L.M.G.C. COSTA², C.H.P. NASCIMENTO³and L. PATUZZI⁴

Received on April 9, 2018 / Accepted on December 7, 2018

ABSTRACT. The matching polytope of a graphG, denoted byM(G), is the convex hull of the set of the incidence vectors of the matchings ofG. The graphG(M(G)), whose vertices and edges are the vertices and edges ofM(G), is the skeleton of the matching polytope ofG. In this paper, for an arbitrary graph, we prove that the minimum degree ofG(M(G))is equal to the number of edges ofG, generalizing a known result for trees. From this, we identify the vertices of the skeleton with the minimum degree and we prove that the union of stars and triangles characterizes regular skeletons of the matching polytopes of graphs.

Keywords: regular graph, matching polytope, degree of matching.

1 INTRODUCTION

LetG=G(V,E)be a simple graph with vertex setV ={v₁,v₂, . . . ,v_n} and set of edgesE = {e₁,e₂, . . . ,e_m}. For eachk,1≤k≤m,e_k=v_iv_jis anincidentedge to theadjacentverticesv_i andvj,1≤i< j≤n. The set of adjacent vertices ofviisNG(v_i), called theneighborhoodofvi, whose cardinalityd(v_i)is thedegreeofv_i. A verticev_i∈V is said to bependantifd(v_i) =1. An edge ofGis said to bependantif one of its vertices has only one neighbor. For a given edgee_k, the set of adjacent edges ofe_kis denotedI(e_k).

Two non adjacent edges aredisjointand a set of pairwise disjoint edgesM is amatchingofG.

An unitary edge set is aone-edge matchingand the empty set is theempty matching,∅. A vertex v∈V is said to beM-saturatedif there is an edge ofMincident tov. Otherwise,vis said to be an M-unsaturatedvertex. Aperfect matching Mis one for which every vertex ofGisM-saturated.

For a natural numberk, apathwith lengthk,P_k+1(or simplyP), is a sequence of distinct vertices v₁v₂. . .v_kv_k+1such that, for 1≤i≤k,e_i=v_iv_i+1is an edge ofG. Acyclewith lengthk,C_k(or

*Corresponding author: Carlos Henrique Pereira do Nascimento – E-mail: carloshenrique@id.uff.br –https://orcid.

org/0000-0001-6023-2397

1Engenharia de Produc¸˜ao, COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brasil. E-mail:

nairabreunovoa@gmail.com.

2Col´egio Pedro II, Rio de Janeiro, Brasil. E-mail: lmgccosta@gmail.com.

3Departamento de Matem´atica, Instituto de Ciˆencias Exatas, Universidade Federal Fluminense, Volta Redonda, Brasil.

E-mail: carloshenrique@id.uff.br.

4Instituto de Matem´atica, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brasil.

simplyC), is obtained fromP_kby adding the edgev₁v_k. Ifkis odd,C_kis said to be anodd cycle.

Otherwise,C_kis aneven cycle. A path or a cycle can also be denoted by an ordered sequence of their respective edges. The ordering of the edges is given by the sequence of the vertices of the path or the cycle. Given a matchingM inG, a pathPis anM-alternating pathinGiff it contains, alternately, edges fromE\M andM. A cycleCisM-alternatingiffCis an even cycle and it contains alternately edges fromE\M andM. For more basic definitions and notations of graphs, see [2], [5] and, for matchings, see [8].

ApolytopeofRⁿis the convex hullP=convex{x₁, . . . ,x_r}of a finite set of vectorsx₁, . . . ,x_r∈ Rⁿ. Given a polytopeP, the skeletonofP is a graph G(P) whose vertices and edges are, respectively, vertices (faces of dimension 0) and edges (faces of dimension 1) ofP.

For the ordered setEofmedges of a graph,R^Eis the vector space of real-valued vectors indexed by the elements ofEwhose dimension is dim(R^E) =m. ForF⊂E, theincidence vectorofFis defined as follows:

χF(e) =

( 1, ife∈F;

0, otherwise.

In general, we identify each subset of edges with its respective incidence vector. Thematching polytopeof G,M(G), is the convex hull of the incidence vectors of the matchings inG. For more definitions and notations of polytopes, see [7].

Two matchingsM andN are saidadjacent,M∼N, if and only if their correspondent vertices χM ≡M and χN ≡N are adjacent in the skeleton of the matching polytope. Thedegree of a matching M, denotedd(M), is the degree of the correspondent vertex inG(M(G)).

Given two sets A andB, the symmetric difference A∆B is defined by (A∪B)\(A∩B). Next theorems characterize the adjacency of two matchingsM andN by their symmetric difference M∆N.

Theorem 1.([3]) Two distinct matchings M and N of a graph G are adjacent in the matching polytopeM(G)if and only if M∆N is a connected subgraph of G.

Theorem 2.([9]) Two distinct matchings M and N of a graph G are adjacent in the matching polytopeM(G)if and only if M∆N is a path or a cycle (even cycle) in G.

Figure 1 displays the cycleC4 and the skeleton of its matching polytope, G(M(C₄)). Since M₁∆M₂is a path andM₅∆M₆is the cycleC₄,M₁∼M₂andM₅∼M₆inG(M(C₄)). However, M₁6∼M₃, onceM₁∆M₃is a disconnected subgraph of the cycle.

LetT be a tree withnvertices. The acyclic Birkhoff polytopeΩn(T)is the set ofn×ndoubly stochastic matricesA= [a_{i j}]such that the diagonal entries ofAcorrespond to the vertices ofTand each positive entry ofAis either on the diagonal or on a position corresponding to an edge ofT. The matching polytopeM(T)and the acyclic Birkhoff polytopeΩ_n(T)are affinely isomorphic

Figure 1:C₄andG(M(C₄))

[4]. The skeleton of Ω_n(T)was studied in [1] and [6] and, in the sequence, we highlight the following contribution given by them.

Theorem 3. ([1]) If T is a tree with n vertices, the minimum degree of G(M(T)) is n−1.

Moreover, for a matching M of T , d(M) =n−1if and only if M=∅or every edge of M is a pendant edge of T .

In this paper, we generalize the above result forG, where Gis an arbitrary graph. Based on it, we prove two theorems of characterization: the first identifies the matchings ofGwith the minimum degree and the second gives a necessary and sufficient condition aboutGin order to haveG(M(G))as a regular graph.

2 THE DEGREE OF A MATCHING

In the present section we generalize the results from Abreu et al. [1]. From Theorem 2,d(∅) =m inG(M(G)). In fact, there aremone-edge matchings, thus∅as exactlymneighbors. On the other hand, letM6=∅be a matching ofGandebe an edge ofG. Ifeis an edge ofMthenMis adjacent to the matchingN=M\{e}. Ifeis not an edge ofMthenMis adjacent to the matching K= (M\{f: f is an edge inMadjacent toe})∪ {e}. Therefore, the degree ofMis greater than or equal to the number of edges.

LetGbe a graph with a matchingMandPbe anM-alternating path with at least two vertices.

We say that:

(i)Pis anoo-M-pathif its pendant edges belong toM;

(ii)Pis acc-M-pathif its pendant vertices are bothM-unsaturated;

(iii)Pis anoc-M-pathif one of its pendant edges belongs toMand one of its pendant vertex is M-unsaturated.

AnM-alternating pathPis called anM-good pathwhenP is one of the paths defined above.

Moreover, ifCis a cycle ofG, thenCis said to be anM-good cyclewhenCis anM-alternating cycle. In this case,M∩E(C)is a perfect matching ofC.

In Figure 2,M={e₁,e3,e6}is a perfect matching of the graph. BecauseMis a perfect matching, there is no unsaturated vertex. Therefore, there is nocc-M-good path noroc-M-good path inG.

However,e₁e₂e₃is anoo-M-good path ofGbute₁e₂is not anM-good path. Moreover,e₁e₂e₃e₄ is anM-good cycle.

Figure 2: The matchingM={e₁,e₃,e₆}and theM-good cycleC=e₁e₂e₃e₄.

If the symmetric differenceM∆N is a path, whereMandNare any two matchings, necessarily it is anM-good path. Note also that whenM∆Nis a cycle then it isM-good and forN₁6=N₂we haveM∆N₁6=M∆N₂. Based on this, the next theorem is a direct consequence from Theorems 1 and 2.

Theorem 4.Let M be a matching of a graph G. The degree of M in the skeletonG(M(G))is given by the sum of the number of paths and cycles M-good.

A matchingMis said to be astrict matching when two edges of M have no a common incident edge,i.e.,

∀e,f ∈M:e6=f =⇒ I(e)∩I(f) =∅.

In the next theorem, we give a formulae to compute the degree of a strict matchingMthat depends only on degree and neighbors of the verticesM-saturated. Before, we observe that ifMis a strict matching andCis a cycle ofGthenCis a notM-alternating cycle. Also note that ifP is an M-good path thenPhas at most one edge inMand its length is at most 3.

Proposition 5.Let M={e₁, . . . ,e_s}be a strict matching of a graph G= (V,E)and let e_i=u_iv_i where ui,vi∈V for each1≤i≤s. The degree of M is

d(M) =k+

s

∑

i=1

(d(u_i)d(v_i)− |N(u_i)∩N(v_i)|),

where k≥0is the number of edges that have no vertex in common with any edge of M.

Proof.LetMbe as in the statement of Proposition 5. In this case,Gdoes not haveM-good cycles

(1)Pis anoo-M-path. Then,P=esuch thate∈M. There aresof these paths inG;

(2)Pis anoc-M-path. So,P=e f such thate∈Mand f∈/M. Of course, for somei, 1≤i≤s, we havee=u_iv_iandf is incident tou_ior tov_i. There are(d(u_i)−1) + (d(v_i)−1)of these paths;

(3) IfPis acc-M-path, we have to consider two possibilities forP. Firstly,P=f egwithe∈M andf,g∈/M. So, for somei, 1≤i≤s,e=u_iv_isuch thatf is incident tou_iandgis incident tov_i. In this case, there are(d(u_i)−1)·(d(v_i)−1)− |N(u_i)∩N(v_i)|of such paths. The second possibility isP= f with f ∈/M. Since Pis acc-M-path, for eache∈E that is incident to f,e∈/ M. The number of such edges is equal to the numberkdefined in the statement of Proposition 5.

From the itens (1), (2) and (3), we obtain d(M) =s+k+

s i=1

∑

((d(u_i)−1) + (d(v_i)−1) + (d(u_i)−1)(d(v_i)−1)− |N(u_i)∩N(v_i)|) =

=k+

s

∑

i=1

(d(u_i)d(v_i)− |N(u_i)∩N(v_i)|).

Theorem 6.Let M and N be matchings of a graph G. If M⊂N, then d(M)≤d(N).

Proof. LetMandNbe matchings of a graphGsuch thatN=M∪ {e}, wheree∈E. Consider BM the sets ofM-good paths andM-good cycles ofG. Similarly, defineBN. From Theorem 4, d(M) =|BM| andd(N) =|BM|. Build the function ϕ_e:BM →BN such that, for every cycle C∈BM,ϕe(C) =Cand, for every pathP∈BM,

ϕe(P) =

( P, ifV(e)∩V(P) =∅;

P∪ {e}, ifV(e)∩V(P) =u, whereuis one of the endpoints ofP.

In fact, sincePis anM-good path,M∪ {e}is a matching ande6∈M, then there are only two possible cases: eitherV(e)∩V(P) =∅orV(e)∩V(P) =u(and therefore isMunsaturated). By construction, for distinct paths or cycles belonging toBM, we have distinct images inBN. Then, ϕ_eis an injective function and so,d(M)≤d(N).

In the general case, letN\M={e₁,e₂, . . . ,e_k},N₁=M∪{e₁},N₂=M∪{e₁,e₂}, . . . ,andN_k=N.

By the same argument used before, we getd(M)≤d(N₁)≤d(N₂)≤ · · · ≤d(N_k−1)≤d(N).

From the previous theorem applied toM=∅, we can also obtain that ifGis a graph with m edges then the minimum degree ofG(M(G))is equal tom.

3 REGULAR MATCHINGS POLYTOPES

In this section we characterize the graphs for which the skeletons are regular and all matchings of a graph with minimum degree.

An edgee=uvof a graphGis called abindifd(u) =d(v) =2 and|N(u)∩N(v)|=1. Note that ifeis a bind ofG,eis an edge of a triangle of graph. However, the reciprocal is not necessarily true.

Proposition 7.Let G be a graph with m edges and M={e}be a one-edge matching of G. The degree of M is d(M) =m if and only if e is either a bind or a pendent edge of G.

Proof.LetGbe a graph withmedges ande=uvan edge ofG. We know thatm−d(u)−d(v) +1 is the number of edges that neither is incident to u nor to v. From Proposition 5, d({e}) = m−d(u)−d(v) +1+d(u)d(v)−t, wheret=|N(u)∩N(v)|. Therefore,d({e}) =mif and only ifd(u)d(v)−t=d(u) +d(v)−1, i.e.,(d(u)−1)(d(v)−1) =t. Sinced(u)>t andd(v)>t, (d(u)−1)(d(v)−1)≥t². So,(d(u)−1)(d(v)−1) =tif and only ift=0 ort=1. In the first case,eis a pendant edge and, otherwise,d(u) =d(v) =2 and so,eis a bind.

Note that, ifGis a graph without binds and pendant edges, the unique vertex ofG(M(G))with the minimum degree isM=∅. It is not difficult to see that, exceptK₃, all 2-connected graphs satisfy this property.

Proposition 8.Let G be a graph with m edges and M6=∅be a matching of G. Then, d(M) =m if and only if M has only pendant edges or binds.

Proof. Suppose there is e∈M such that e is neither a bind nor a pendant edge ofG. So, from Proposition 7 and Theorem 6, m<d({e})≤d(M). Consequently, m6=d(M). By the contrapositive, ifd(M) =m, every edge ofMis a pendant edge or a bind of the graph.

Suppose now that M is a matching of Gwiths edges such that ife∈M,eis a bind or eis a pendant edge ofG. LetN be a matching such thatN∼M inG(M(G)). From here and by Theorem 2,M∆N is anM-good path Por anM-good cycleCof G. SinceCis an even cycle, C6=K3. So,Cdoes not have binds. Consequently,M∆Nis a path. Concerning the pathP, only its pendant edges can belong toM. Moreover, oncePis an alternated path, it has length at most length 3. Hence, there are only the following possibilities toP:

1. IfPis anoo-M-path, thenP=e, wheree∈M, orP=e₁f e₂, wheref∈/MandI(f)∩M= {e₁,e₂}. In the first case, there ares possibilities to P and, in the second, there aret₁ possibilities toP, wheret₁is the number of edges ofE\Msuch that both terminal vertices are incident to an edge ofM;

2. IfPis acc-M-path,P=f, wheref ∈/MandI(f)∩M=∅. Here, there aret₂possibilities toP,wheret₂is the number of edges ofE\Mfor which any edge ofM does not incident

3. Finally, ifPis anoc-M-path, thenP=e f, where f ∈/MandI(f)∩M={e}. In this last case, there aret₃possibilities toP, wheret₃is the number of edges ofE\M which only one end vertex of edge is incident to some edge ofM.

From (1), (2) and (3) possibilities above, s+ ∑³

i=1

t_i =s+|E\M|=m is the number of the possibilities to havePas an M-good path ofG. From Theorem 4 it follows thatd(M) =m.

Finally, next theorem proves that the union of stars and triangles characterizes regular skeletons of the matching polytopes of graphs.

Theorem 9.Let G be a graph with m edges. The skeletonG(M(G))is an m-regular graph if and only if G is a disjoint union of stars and triangles.

Proof. Suppose thatG(M(G))is anm-regular graph. Then, for everye∈E, we haved({e}) = d(∅) =m. Besides, by Proposition 7, this occurs only ifeis a bind or a pendant edge ofG. So, Gbe a graph that is a disjoint union of stars and triangles.

Suppose now thatGis a disjoint union of stars and triangles. Therefore, any matching ofGhas only pendant edges or binds. From Proposition 8,G(M(G))is anm-regular graph.

Figure 3 displays the graphG=K₃∪S_1,1and its skeleton,G(M(K₃∪S_1,1)).

Figure 3:G=K₃∪S_1,1andG(M(G))

4 FINAL CONSIDERATIONS

In this paper we give a closed formula to compute the degree of a strict matching, i.e., a matching M={e₁, . . . ,e_k}such that ∀i,j∈ {1, . . . ,k}, e_i, e_j∈M, e_i6=e_j ⇒I(e_i)∩I(e_j) = ∅.Abreu

et al. [1] give the minimum degree of the matching polytope for a tree. We generalize this result for any graph. We characterize all the graphsGfor whichG(M(G))is regular. Besides, in the last graph, we find all vertices with minimum degree. Finally, we emphasize that the problem of determination of the maximum degree ofG(M(G))is still unresolved.

5 ACKNOWLEDGMENT

The authors are grateful to anonymous referees for their suggestions and comments that im- proved substantially the quality of this article. This work was partially supported by CNPq, PQ-304177/2013-0 and Universal Project-442241/2014-3, and by the Portuguese Foundation for Science and Technology (FCT), CIDMA-project UID/MAT/04106/2013.

RESUMO. O politopo de emparelhamentos de um grafoG, denotado porM(G), ´e o fecho convexo do conjunto dos vetores de incidˆencia dos emparelhamentos deG. O grafo G(M(G)), cujos v´ertices e arestas s˜ao os v´ertices e arestas deM(G), ´e o esqueleto do politopo de emparelhamentos deG. Neste artigo, para um grafo arbitr´ario, n´os provamos que o grau m´ınimo deG(M(G))´e igual ao n´umero de arestas deG, generalizando um co- nhecido resultado para ´arvores. Al´em disso, n´os identificamos os v´ertices do esqueleto que possuem grau m´ınimo e provamos que a uni˜ao de estrelas e triˆangulos caracteriza esqueletos regulares de politopos de emparelhamentos de grafos.

Palavras-chave:grafo regular, politopo de emparelhamentos, grau de um emparelhamento.

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