ABSTRACT
The matching polytope of a graph G, denoted by ℳ (G), is the convex hull of the set of the incidence vectors of the matchings of G. The graph 𝒢 (ℳ (G)), whose vertices and edges are the vertices and edges of ℳ (G), is the skeleton of the matching polytope of G. In this paper, for an arbitrary graph, we prove that the minimum degree of 𝒢 (ℳ (G)) is equal to the number of edges of G, 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
RESUMO
O politopo de emparelhamentos de um grafo G, denotado por ℳ (G), é o fecho convexo do conjunto dos vetores de incidência dos emparelhamentos de G. O grafo 𝒢 (ℳ (G)), cujos vértices e arestas são os vértices e arestas de ℳ (G), é o esqueleto do politopo de emparelhamentos de G. Neste artigo, para um grafo arbitrário, nós provamos que o grau mínimo de 𝒢 (ℳ (G)) é igual ao número de arestas de G, generalizando um conhecido resultado para árvores. Além disso, nós identificamos os vértices do esqueleto que possuem grau mínimo e provamos que a união de estrelas e triângulos caracteriza esqueletos regulares de politopos de emparelhamentos de grafos.
Palavras-chave:
grafo regular; politopo de emparelhamentos; grau de um emparelhamento
1 INTRODUCTION
Let be a simple graph with vertex set and set of edges . For each is an incident edge to the adjacent vertices v i and v j , . The set of adjacent vertices of v i is N G (v i ), called the neighborhood of v i , whose cardinality d(v i ) is the degree of v i . A vertice is said to be pendant if . An edge of G is said to be pendant if one of its vertices has only one neighbor. For a given edge e k , the set of adjacent edges of e k is denoted I(e k ).
Two non adjacent edges are disjoint and a set of pairwise disjoint edges M is a matching of G. An unitary edge set is a one-edge matching and the empty set is the empty matching, Ø. A vertex is said to be M-saturated if there is an edge of M incident to v. Otherwise, v is said to be an M-unsaturated vertex. A perfect matching M is one for which every vertex of G is M-saturated.
For a natural number k, a path with length k, P k+1 (or simply P), is a sequence of distinct vertices such that, for is an edge of G. A cycle with length k, C k (or simply C), is obtained from P k by adding the edge v 1 v k . If k is odd, C k is said to be an odd cycle. Otherwise, C k is an even 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 matching M in G, a path P is an M-alternating path in G iff it contains, alternately, edges from E\M and M. A cycle C is M-alternating iff C is an even cycle and it contains alternately edges from E\M and M. For more basic definitions and notations of graphs, see 22 B. Bollobas. “Modern Graph Theory”. Graduate Texts in Mathematics. Springer, New York (1998)., 55 R. Diestel. “Graph Theory”. Springer-Verlag, New York (2000). and, for matchings, see 88 L. Lovasz & M.D. Plumer. “Matching Theory”. Ann. Discrete Math. 29 121. North-Holland, Amsterdam (1986)..
A polytope of ℝn is the convex hull of a finite set of vectors . Given a polytope 𝒫, the skeleton of 𝒫 is a graph 𝒢 (𝒫) whose vertices and edges are, respectively, vertices (faces of dimension 0) and edges (faces of dimension 1) of 𝒫.
For the ordered set E of m edges of a graph, ℝE is the vector space of real-valued vectors indexed by the elements of E whose dimension is . For , the incidence vector of F is defined as follows:
In general, we identify each subset of edges with its respective incidence vector. The matching polytope of G, ℳ (G), is the convex hull of the incidence vectors of the matchings in G. For more definitions and notations of polytopes, see 77 B. Grunbaum. “Convex Polytopes”. Springer-Verlag, New York (2003)..
Two matchings M and N are said adjacent, , if and only if their correspondent vertices and are adjacent in the skeleton of the matching polytope. The degree of a matching M, denoted d(M), is the degree of the correspondent vertex in 𝒢 (ℳ (G)).
Given two sets A and B, the symmetric difference AΔB is defined by . Next theorems characterize the adjacency of two matchings M and N by their symmetric difference MΔN.
Theorem 1.(33 V. Chvatal. On certain polytopes associated with graphs. Journal of Combinatorial Theory, B 18 (1975), 138-154.) Two distinct matchings M and N of a graph G are adjacent in the matching polytope ℳ (G) if and only if MΔN is a connected subgraph of G.
Theorem 2.(99 A. Schrijver. “Combinatorial optimization: polyhedra and efficiency”. Algorithms and Combinatorics 24. Springer-Verlag, New York (2003).) Two distinct matchings M and N of a graph G are adjacent in the matching polytope ℳ (G) if and only if MΔN is a path or a cycle (even cycle) in G.
Figure 1 displays the cycle C 4 and the skeleton of its matching polytope, 𝒢 (ℳ (C 4)). Since M 1ΔM 2 is a path and M 5ΔM 6 is the cycle C 4, and in 𝒢 (ℳ (C4)). However, , once M 1ΔM 3 is a disconnected subgraph of the cycle.
Let T be a tree with n vertices. The acyclic Birkhoff polytope Ωn (T) is the set of doubly stochastic matrices such that the diagonal entries of A correspond to the vertices of T and each positive entry of A is either on the diagonal or on a position corresponding to an edge of T. The matching polytope ℳ (T) and the acyclic Birkhoff polytope Ωn (T) are affinely isomorphic 44 L. Costa , C.M. da Fonseca & E.A. Martins. The diameter of the acyclic Birkhoff polytope. Linear Algebra Appl, 428 (2008), 1524-1537.. The skeleton of Ωn (T) was studied in 11 N. Abreu, L. Costa, G. Dahl & E. Martins. The skeleton of acyclic Birkhoff polytopes. Linear Algebra Appl., 457 (2014), 29-48. and 66 R. Fernandes. Computing the degree of a vertex in the skeleton of acyclic Birkhoff polytopes. Linear Algebra Appl, 475 (2015), 119-133. and, in the sequence, we highlight the following contribution given by them.
Theorem 3.(11 N. Abreu, L. Costa, G. Dahl & E. Martins. The skeleton of acyclic Birkhoff polytopes. Linear Algebra Appl., 457 (2014), 29-48.) If T is a tree with n vertices, the minimum degree of 𝒢 (ℳ (T)) is. Moreover, for a matching M of T,if and only ifor every edge of M is a pendant edge of T.
In this paper, we generalize the above result for G, where G is an arbitrary graph. Based on it, we prove two theorems of characterization: the first identifies the matchings of G with the minimum degree and the second gives a necessary and sufficient condition about G in order to have 𝒢 (ℳ (G)) as a regular graph.
2 THE DEGREE OF A MATCHING
In the present section we generalize the results from Abreu et al. 11 N. Abreu, L. Costa, G. Dahl & E. Martins. The skeleton of acyclic Birkhoff polytopes. Linear Algebra Appl., 457 (2014), 29-48.. From Theorem 2, in 𝒢 (ℳ (G)). In fact, there are m one-edge matchings, thus Ø as exactly m neighbors. On the other hand, let a matching of G and e be an edge of G. If e is an edge of M then M is adjacent to the matching . If e is not an edge of M then M is adjacent to the matching . Therefore, the degree of M is greater than or equal to the number of edges.
Let G be a graph with a matching M and P be an M-alternating path with at least two vertices. We say that:
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(i) P is an oo-M-path if its pendant edges belong to M;
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(ii) P is a cc-M-path if its pendant vertices are both M-unsaturated;
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(iii) P is an oc-M-path if one of its pendant edges belongs to M and one of its pendant vertex is M-unsaturated.
An M-alternating path P is called an M-good path when P is one of the paths defined above. Moreover, if C is a cycle of G, then C is said to be an M-good cycle when C is an M-alternating cycle. In this case, is a perfect matching of C.
In Figure 2, is a perfect matching of the graph. Because M is a perfect matching, there is no unsaturated vertex. Therefore, there is no cc-M-good path nor oc-M-good path in G. However, e 1 e 2 e 3 is an oo-M-good path of G but e 1 e 2 is not an M-good path. Moreover, e 1 e 2 e 3 e 4 is an M-good cycle.
If the symmetric difference MΔN is a path, where M and N are any two matchings, necessarily it is an M-good path. Note also that when MΔN is a cycle then it is M-good and for we have . 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 skeleton 𝒢 (ℳ (G)) is given by the sum of the number of paths and cycles M-good.
A matching M is said to be a strict matching when two edges of M have no a common incident edge, i.e.,
In the next theorem, we give a formulae to compute the degree of a strict matching M that depends only on degree and neighbors of the vertices M-saturated. Before, we observe that if M is a strict matching and C is a cycle of G then C is a not M-alternating cycle. Also note that if P is an M-good path then P has at most one edge in M and its length is at most 3.
Proposition 5.Letbe a strict matching of a graphand letwherefor each. The degree of M is
where is the number of edges that have no vertex in common with any edge of M.
Proof. Let M be as in the statement of Proposition 5. In this case, G does not have M-good cycles and, if P is an M-good path, P has at most length 3 with at most one edge of M. Hence, if P is an M-good path of G, P has to satisfy one of the cases below.
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(1) P is an oo-M-path. Then, such that . There are s of these paths in G;
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(2) P is an oc-M-path. So, such that and . Of course, for some , we have and f is incident to u i or to v i . There are of these paths;
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(3) If P is a cc-M-path, we have to consider two possibilities for P. Firstly, with and . So, for some such that f is incident to u i and g is incident to v i . In this case, there are of such paths. The second possibility is with .Since P is a cc-M-path, for each that is incident to . The number of such edges is equal to the number k defined in the statement of Proposition 5.
From the itens (1), (2) and (3), we obtain
◻
Theorem 6. Let M and N be matchings of a graph G. If , then .
Proof. Let M and N be matchings of a graph G such that , where . Consider ??M the sets of M-good paths and M-good cycles of G. Similarly, define 𝔹N . From Theorem 4, and . Build the function such that, for every cycle and, for every path ,
In fact, since P is an M-good path, is a matching and , then there are only two possible cases: either or (and therefore is M unsaturated). By construction, for distinct paths or cycles belonging to 𝔹M , we have distinct images in 𝔹N . Then, φ e is an injective function and so, .
In the general case, let , and . By the same argument used before, we get .
From the previous theorem applied to , we can also obtain that if G is a graph with m edges then the minimum degree of 𝒢 (ℳ (G)) is equal to m.
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 edge of a graph G is called a bind if and . Note that if e is a bind of G, e is 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 be a one-edge matching of G. The degree of M is if and only if e is either a bind or a pendent edge of G.
Proof. Let G be a graph with m edges and an edge of G.We know that is the number of edges that neither is incident to u nor to v. From Proposition 5, , where . Therefore, if and only if . Since and , . So, if and only if or . In the first case, e is a pendant edge and, otherwise, and so, e is a bind.◻
Note that, if G is a graph without binds and pendant edges, the unique vertex of 𝒢 (ℳ (G)) with the minimum degree is . It is not difficult to see that, except K 3, all 2-connected graphs satisfy this property.
Proposition 8. Let G be a graph with m edges and be a matching of G. Then, if and only if M has only pendant edges or binds.
Proof. Suppose there is such that e is neither a bind nor a pendant edge of G. So, from Proposition 7 and Theorem 6, . Consequently, . By the contrapositive, if , every edge of M is a pendant edge or a bind of the graph.
Suppose now that M is a matching of G with s edges such that if , e is a bind or e is a pendant edge of G. Let N be a matching such that in 𝒢 (ℳ (G)). From here and by Theorem 2, MΔN is an M-good path P or an M-good cycle C of G. Since C is an even cycle, . So, C does not have binds. Consequently, MΔN is a path. Concerning the path P, only its pendant edges can belong to M. Moreover, once P is an alternated path, it has length at most length 3. Hence, there are only the following possibilities to P:
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If P is an oo-M-path, then , where , or , where and . In the first case, there are s possibilities to P and, in the second, there are t 1 possibilities to P, where t 1 is the number of edges of E\M such that both terminal vertices are incident to an edge of M;
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If P is a cc-M-path, , where and . Here, there are t 2 possibilities to P; where t 2 is the number of edges of E\M for which any edge of M does not incident to the end vertices of those edges;
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Finally, if P is an oc-M-path, then , where and . In this last case, there are t 3 possibilities to P, where t 3 is the number of edges of E\M which only one end vertex of edge is incident to some edge of M.
From (1), (2) and (3) possibilities above, is the number of the possibilities to have P as an M-good path of G. From Theorem 4 it follows that . ◻
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 skeleton 𝒢 (ℳ (G)) is an m-regular graph if and only if G is a disjoint union of stars and triangles.
Proof. Suppose that 𝒢 (ℳ (G)) is an m-regular graph. Then, for every , we have . Besides, by Proposition 7, this occurs only if e is a bind or a pendant edge of G. So, G be a graph that is a disjoint union of stars and triangles.
Suppose now that G is a disjoint union of stars and triangles. Therefore, any matching of G has only pendant edges or binds. From Proposition 8, 𝒢 (ℳ (G)) is an m-regular graph. ◻
Figure 3 displays the graph and its skeleton, .
4 FINAL CONSIDERATIONS
In this paper we give a closed formula to compute the degree of a strict matching, i.e., a matching such that . Abreu et al. 11 N. Abreu, L. Costa, G. Dahl & E. Martins. The skeleton of acyclic Birkhoff polytopes. Linear Algebra Appl., 457 (2014), 29-48. give the minimum degree of the matching polytope for a tree. We generalize this result for any graph. We characterize all the graphs G for which 𝒢 (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 of 𝒢 (ℳ (G)) is still unresolved.
5 ACKNOWLEDGMENT
The authors are grateful to anonymous referees for their suggestions and comments that improved 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.
REFERENCES
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1N. Abreu, L. Costa, G. Dahl & E. Martins. The skeleton of acyclic Birkhoff polytopes. Linear Algebra Appl., 457 (2014), 29-48.
-
2B. Bollobas. “Modern Graph Theory”. Graduate Texts in Mathematics. Springer, New York (1998).
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3V. Chvatal. On certain polytopes associated with graphs. Journal of Combinatorial Theory, B 18 (1975), 138-154.
-
4L. Costa , C.M. da Fonseca & E.A. Martins. The diameter of the acyclic Birkhoff polytope. Linear Algebra Appl, 428 (2008), 1524-1537.
-
5R. Diestel. “Graph Theory”. Springer-Verlag, New York (2000).
-
6R. Fernandes. Computing the degree of a vertex in the skeleton of acyclic Birkhoff polytopes. Linear Algebra Appl, 475 (2015), 119-133.
-
7B. Grunbaum. “Convex Polytopes”. Springer-Verlag, New York (2003).
-
8L. Lovasz & M.D. Plumer. “Matching Theory”. Ann. Discrete Math. 29 121. North-Holland, Amsterdam (1986).
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9A. Schrijver. “Combinatorial optimization: polyhedra and efficiency”. Algorithms and Combinatorics 24. Springer-Verlag, New York (2003).
Publication Dates
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Publication in this collection
10 June 2019 -
Date of issue
Jan-Apr 2019
History
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Received
09 Apr 2018 -
Accepted
07 Dec 2018