Learn more in our Algorithm Fundamentals course, built by experts for you. 8-12, 1974. Graph 1Graph\ 1Graph 1 shows all the edges, in blue, that connect the bipartite graph. This added complexity often stems from graph labeling, where edges or vertices labeled with quantitative attributes, such as weights, costs, preferences or any other specifications, which adds constraints to potential matches. You run it on a graph and a matching, and it returns a path. The augmenting path algorithm is a pain, but I'll describe it below. That is, every vertex of the graph is incident to exactly one edge of the matching. Cambridge, Sumner, D. P. "Graphs with 1-Factors." cubic graph with 0, 1, or 2 bridges Try to draw out the alternating path and see what vertices the path starts and ends at. [4], The blossom algorithm works by running the Hungarian algorithm until it runs into a blossom, which it then shrinks down into a single vertex. S is a perfect matching if every vertex is matched. At the end, a perfect matching is obtained. matchings are only possible on graphs with an even number of vertices. and the corresponding numbers of connected simple graphs are 1, 5, 95, 10297, ... The nine perfect matchings of the cubical graph In this paper, we determine graph isomorphism with the help of perfect matching algorithm, to limit the range of search of 1 to 1 correspondences between the two graphs: We reconfigure the graphs into layered graphs, labeling vertices by partitioning the set of vertices by degrees. In many of these applications an artificial society of agents, usually representing humans or animals, is created, and the agents need to be paired with each other to allow for interactions between them. Matching algorithms are algorithms used to solve graph matching problems in graph theory. A perfect matching of a graph is a matching (i.e., an independent edge set) in which every vertex of the graph is incident to exactly one edge of the matching. biology (Gras et al. The general procedure used begins with finding any maximal matching greedily, then expanding the matching using augmenting paths via almost augmenting paths. The algorithm is taken from "Efficient Algorithms for Finding Maximum Matching in Graphs" by Zvi Galil, ACM Computing Surveys, 1986. More formally, the algorithm works by attempting to build off of the current matching, MMM, aiming to find a larger matching via augmenting paths. It is based on the "blossom" method for finding augmenting paths and the "primal-dual" method for finding a matching of maximum weight, both due to Jack Edmonds. §VII.5 in CRC Handbook of Combinatorial Designs, 2nd ed. Finding augmenting paths in a graph signals the lack of a maximum matching. We distinguish the cases p even and p odd.. For p even, the complete bipartite graph K p/2,p/2 is a union of p /2 edge-disjoint perfect matchings (if the vertices are x 0, …, x p/2-1 and y 0, …, y p/2-1, then the i-th matching joins x j with y j+1 with indices modulo p/2). Shrinking of a cycle using the blossom algorithm. 740-755, A common bipartite graph matching algorithm is the Hungarian maximum matching algorithm, which finds a maximum matching by finding augmenting paths. A perfect Augmenting paths in matching problems are closely related to augmenting paths in maximum flow problems, such as the max-flow min-cut algorithm, as both signal sub-optimality and space for further refinement. 193-200, 1891. The #1 tool for creating Demonstrations and anything technical. In max-flow problems, like in matching problems, augmenting paths are paths where the amount of flow between the source and sink can be increased. a,b,d and e are included in no perfect matching, and c and f are included in all the perfect matchings. While not all graphs have a perfect matching, all graphs do have a maximum independent edge set (i.e., a maximum matching; Skiena 1990, p. 240; Pemmaraju An example of a matching is [{m1,w1},{m2,w2},{m3,w3}] (m4 is unmatched) In the example you gave a possible matching can be a perfect matching because every member of M can be matched uniquely to a member of W. Introduction to Graph Theory, 2nd ed. 29 and 343). The poor performance of the Hungarian Matching Algorithm sometimes deems it unuseful in dense graphs, such as a social network. Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. (OEIS A218463). The goal of a matching algorithm, in this and all bipartite graph cases, is to maximize the number of connections between vertices in subset AAA, above, to the vertices in subset BBB, below. Weisstein, Eric W. "Perfect Matching." Sign up to read all wikis and quizzes in math, science, and engineering topics. An augmenting path, then, builds up on the definition of an alternating path to describe a path whose endpoints, the vertices at the start and the end of the path, are free, or unmatched, vertices; vertices not included in the matching. This implies that the matching MMM is a maximum matching. Graph matching algorithms often use specific properties in order to identify sub-optimal areas in a matching, where improvements can be made to reach a desired goal. Math. I'm aware of (some) of the literature on this topic, but as a non-computer scientist I'd rather not have to twist my mind around one of the Blossum algorithms. Abstract. Forgot password? Knowledge-based programming for everyone. p. 344). If a graph has a Hamiltonian cycle, it has two different perfect matchings, since the edges in the cycle could be alternately colored. Prove that in a tree there is at most $1$ perfect matching. Proc. New York: Springer-Verlag, 2001. Pemmaraju, S. and Skiena, S. Computational Discrete Mathematics: Combinatorics and Graph Theory in Mathematica. Join the initiative for modernizing math education. How to make a computer do what you want, elegantly and efficiently. "Claw-Free Graphs--A 2002), economics (Deissenberg et al. Tutte, W. T. "The Factorization of Linear Graphs." Unmatched bipartite graph. In an unweighted graph, every perfect matching is a maximum matching and is, therefore, a maximal matching as well. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. … From MathWorld--A Wolfram Web Resource. Reading, A perfect matching is therefore a matching containing $n/2$ edges (the largest possible), meaning perfect matchings are only possible on graphs with an even number of vertices. The numbers of simple graphs on , 4, 6, ... vertices Boca Raton, FL: CRC Press, pp. Shrinking of a cycle using the blossom algorithm. A perfect matching is therefore a matching containing edges (the largest possible), meaning perfect matchings are only possible on graphs with an even number of vertices. Petersen's theorem states that every cubic graph with no bridges has a perfect matching (Petersen Unfortunately, not all graphs are solvable by the Hungarian Matching algorithm as a graph may contain cycles that create infinite alternating paths. A graph has a perfect matching iff Lovász, L. and Plummer, M. D. Matching Proof. A perfect matching is therefore a matching containing From online matchmaking and dating sites, to medical residency placement programs, matching algorithms are used in areas spanning scheduling, planning, pairing of vertices, and network flows. Today we extend Edmond’s matching algorithm to weighted graphs. Each time an augmenting path is found, the number of matches, or total weight, increases by 1. matching). West, D. B. More specifically, matching strategies are very useful in flow network algorithms such as the Ford-Fulkerson algorithm and the Edmonds-Karp algorithm. has no perfect matching iff there is a set whose Acta Math. 15, For a detailed explanation of the concepts involved, see Maximum_Matchings.pdf. In practice, researchers have found that Hopcroft-Karp is not as good as the theory suggests — it is often outperformed by breadth-first and depth-first approaches to finding augmenting paths.[1]. has a perfect matching.". matching is sometimes called a complete matching or 1-factor. In mathematics, economics, and computer science, the stable marriage problem (also stable matching problem or SMP) is the problem of finding a stable matching between two equally sized sets of elements given an ordering of preferences for each element. The theoreticians have proven that this works. Bipartite matching is used, for example, to match men and women on a dating site. We also show a sequential implementation of our algo- rithmworkingin Every perfect matching is a maximum matching but not every maximum matching is a perfect matching. If the search finds an augmenting path, the matching gains one more edge. any edge of Trim(G) is incident to no edge of M \ Trim(M),M∪ (M \ Trim(M)) isincluded in M(G)foranyM ∈M(IS(Trim(G))). Computational Discrete Mathematics: Combinatorics and Graph Theory in Mathematica. Explore anything with the first computational knowledge engine. [6]. Once the matching is updated, the algorithm continues and searches again for a new augmenting path. Godsil, C. and Royle, G. Algebraic A common characteristic investigated within a labeled graph is a known as feasible labeling, where the label, or weight assigned to an edge, never surpasses in value to the addition of respective vertices’ weights. A perfect matching is also a minimum-size edge cover (from wiki). The time complexity of this algorithm is O(∣E∣∣V∣)O(|E| \sqrt{|V|})O(∣E∣∣V∣) in the worst case scenario, for ∣E∣|E|∣E∣ total edges and ∣V∣|V|∣V∣ total vertices found in the graph. The graph does contain an alternating path, represented by the alternating colors below. Englewood Cliffs, NJ: Prentice-Hall, pp. A graph If you consider a graph with 4 vertices connected so that the graph resembles a square, there are two perfect matching sets, which are the pairs of parallel edges. 8v2V x( (v)) = 1 8UˆV;jUj= odd x( (U)) 1 8e2E x e 0 But this program has exponentially-many constraints. Note: The term comes from matching each vertex with exactly one other vertex. This application demonstrates an algorithm for finding maximum matchings in bipartite graphs. Then, it begins the Hungarian algorithm again. 2009), sociology (Macy et al. Or a Python interface to one? After Douglas Bass (dbass@stthomas.edu) 5 Sep 1999. Graph matching problems generally consist of making connections within graphs using edges that do not share common vertices, such as pairing students in a class according to their respective qualifications; or it may consist of creating a bipartite matching, where two subsets of vertices are distinguished and each vertex in one subgroup must be matched to a vertex in another subgroup. Sloane, N. J. All alphabets of patterns must be matched to corresponding matched subsequence. Practice online or make a printable study sheet. This algorithm, known as the hungarian method, is … edges (the largest possible), meaning perfect It's nicer to use than a bipartite matching algorithm on all possible bipartitions, and will always find a minimal perfect matching in the TSP case. 1.1 Technical ideas Our main new technical idea is that of a matching-mimicking network. https://mathworld.wolfram.com/PerfectMatching.html. 2011). Linear-programming duality provides a stopping rule used by the algorithm to verify the optimality of a proposed solution. Note that d ⩽ p − 1 by assumption. CRC Handbook of Combinatorial Designs, 2nd ed. Math. The algorithm was later improved to O(∣V∣3)O(|V|^3)O(∣V∣3) time using better performing data structures. Walk through homework problems step-by-step from beginning to end. The majority of realistic matching problems are much more complex than those presented above. Graph Theory. de Recherche Opér. A perfect matching(also called 1-factor) is a matching in which every node is matched, thus its size We know polynomial-time algorithms to find perfect matchings in graphs. Since every vertex has to be included in a perfect matching, the number of edges in the matching must be where V is the number of vertices. Andersen, L. D. "Factorizations of Graphs." Where l(x)l(x)l(x) is the label of xxx, w(x,y)w(x,y)w(x,y) is the weight of the edge between xxx and yyy, XXX is the set of nodes on one side of the bipartite graph, and YYY is the set of nodes on the other side of the graph. of vertices is missed by a matching that covers all remaining vertices (Godsil and A. Sequences A218462 If there is a feasible labeling within items in MMM, and MMM is a perfect matching, then MMM is a maximum-weight matching. In this specific scenario, the blossom algorithm can be utilized to find a maximum matching. and A218463. perfect matching NC algorithm of [1]. Computation. Perfect Matching – A matching of graph is said to be perfect if every vertex is connected to exactly one edge. A perfect matching of a graph is a matching (i.e., an independent edge set) in which every vertex of the graph is incident to exactly one edge of the matching. having a perfect matching are 1, 6, 101, 10413, ..., (OEIS A218462), Microsimulations and agent-based models (ABMs) are increasingly used across a broad area of disciplines, e.g. A blossom is a cycle in GGG consisting of 2k+12k + 12k+1 edges of which exactly kkk belong to MMM, and where one of the vertices, vvv, the base, in the cycle is at the head of an alternating path of even length, the path being named stem, to an exposed vertex, www[3]. Hopcroft-Karp works by repeatedly increasing the size of a partial matching via augmenting paths. vertex-transitive graph on an odd number To able to solve this problem, vertex count must be even. "Die Theorie der Regulären Graphen." 1891; Skiena 1990, p. 244). of the graph is incident to exactly one edge of the matching. If another blossom is found, it shrinks the blossom and starts the Hungarian algorithm yet again, and so on until no more augmenting paths or cycles are found. Vergnas 1975). However, a number of ideas are needed to find such a cut in NC; the central one being an NC algorithm for finding a face of the perfect matching polytope at which $\Omega(n)$ new conditions, involving constraints of the polytope, are simultaneously satisfied. A parallel algorithm is one where we allow use of polynomially many processors running in parallel. and Skiena 2003, pp. It then constructs a tree using a breadth-first search in order to find an augmenting path. [1]. For the other case can you apply induction using $2$ leaves ? Dordrecht, Netherlands: Kluwer, 1997. Soc. Exact string matching algorithms is to find one, several, or all occurrences of a defined string (pattern) in a large string (text or sequences) such that each matching is perfect. The input to each phase is a pseudo perfect matching and the output of each phase is a new pseudo perfect matching, with number of 3-degree vertices in it, reduced by a constant factor. https://mathworld.wolfram.com/PerfectMatching.html. 22, 107-111, 1947. A variety of other graph labeling problems, and respective solutions, exist for specific configurations of graphs and labels; problems such as graceful labeling, harmonious labeling, lucky-labeling, or even the famous graph coloring problem. Improving upon the Hungarian Matching algorithm is the Hopcroft–Karp algorithm, which takes a bipartite graph, G(E,V)G(E,V)G(E,V), and outputs a maximum matching. Recall that a matchingin a graph is a subset of edges in which every vertex is adjacent to at most one edge from the subset. Zinn (2012) addresses some of the conceptual challenges of findi… Equality graphs are helpful in order to solve problems by parts, as these can be found in subgraphs of the graph GGG, and lead one to the total maximum-weight matching within a graph. 17, 257-260, 1975. 2008) and epidemiology (Gray et al. admits a matching saturating A, which is a perfect matching. Conversely, if the labeling within MMM is feasible and MMM is a maximum-weight matching, then MMM is a perfect matching. . Notes: We’re given A and B so we don’t have to nd them. In Annals of Discrete Mathematics, 1995. 42, Log in here. Using the same method as in the second proof of Hall’s Theorem, we give an algorithm which, given a bipartite graph ((A,B),E) computes either a matching saturating A or a set S such that |N(S)| < |S|. Graph 1Graph\ 1Graph 1, with the matching, MMM, is said to have an alternating path if there is a path whose edges are in the matching, MMM, and not in the matching, in an alternating fashion. A perfect matching is a matching which matches all vertices of the graph. Furthermore, every perfect matching is a maximum independent edge set. Unlike the Hungarian Matching Algorithm, which finds one augmenting path and increases the maximum weight by of the matching by 111 on each iteration, the Hopcroft-Karp algorithm finds a maximal set of shortest augmenting paths during each iteration, allowing it to increase the maximum weight of the matching with increments larger than 111. Graph 1Graph\ 1Graph 1. Amsterdam, Netherlands: Elsevier, 1986. If the number of vertices is even$\implies$ number of edges odd, not divisible by $2$, so no perfect matching. An alternating path usually starts with an unmatched vertex and terminates once it cannot append another edge to the tail of the path while maintaining the alternating sequence. The graph illustrated above is 16-node graph with no perfect matching that is implemented in the Wolfram Language as GraphData["NoPerfectMatchingGraph"]. a e f b c d Fig.2. Maximum is not … https://en.wikipedia.org/wiki/Hopcroft%E2%80%93Karp_algorithm, http://demonstrations.wolfram.com/TheHungarianMaximumMatchingAlgorithm/, https://en.wikipedia.org/wiki/Blossom_algorithm, https://en.wikipedia.org/wiki/File:Edmonds_blossom.svg, http://matthewkusner.com/MatthewKusner_BlossomAlgorithmReport.pdf, http://stanford.edu/~rezab/dao/projects_reports/shoemaker_vare.pdf, https://brilliant.org/wiki/matching-algorithms/. An alternating path in Graph 1 is represented by red edges, in MMM, joined with green edges, not in MMM. If the search is unsuccessful, the algorithm terminates as the current matching must be the largest-size matching possible.[2]. perfect matching algorithm? Faudree, R.; Flandrin, E.; and Ryjáček, Z. Maximum Bipartite Matching Maximum Bipartite Matching Given a bipartite graph G = (A [B;E), nd an S A B that is a matching and is as large as possible. If there exists an augmenting path, ppp, in a matching, MMM, then MMM is not a maximum matching. Two famous properties are called augmenting paths and alternating paths, which are used to quickly determine whether a graph contains a maximum, or minimum, matching, or the matching can be further improved. 164, 87-147, 1997. Precomputed graphs having a perfect matching return True for GraphData[g, "PerfectMatching"] in the Wolfram Hints help you try the next step on your own. The Hopcroft-Karp algorithm uses techniques similar to those used in the Hungarian algorithm and the Edmonds’ blossom algorithm. A matching is a bijection from the elements of one set to the elements of the other set. This problem has various algorithms for different classes of graphs. The main idea is to augment MMM by the shortest augmenting path making sure that no constraints are violated. Bold lines are edges of M.Arcs a,b,c,d,e and f are included in no directed cycle. England: Cambridge University Press, 2003. There is no perfect match possible because at least one member of M cannot be matched to a member of W, but there is a matching possible. Already have an account? set and is the edge set) Perfect matching was also one of the first problems to be studied from the perspective of parallel algorithms. Soc. Deciding whether a graph admits a perfect matching can be done in polynomial time, using any algorithm for finding a maximum cardinality matching.. l(x)+l(y)≥w(x,y),∀x∈X, ∀y∈Yl(x) + l(y) \geq w(x,y), \forall x \in X,\ \forall y \in Yl(x)+l(y)≥w(x,y),∀x∈X, ∀y∈Y. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. An alternating path in Graph 1 is represented by red edges, in. So, the challenging part is finding an augmenting path. Once the path is built from B1B1B1 to node A5A5A5, no more red edges, edges in MMM, can be added to the alternating path, implying termination. An equality graph for a graph G=(V,Et)G = (V, E_t)G=(V,Et) contains the following constraint for all edges in a matching: El={(x,y)}:l(x)+l(y)=w(x,y)}E_l = \{(x,y)\} : l(x) + l(y) = w(x,y)\}El={(x,y)}:l(x)+l(y)=w(x,y)}. Note that rather confusingly, the class of graphs known as perfect Does the matching in this graph have an augmenting path, or is it a maximum matching? No polynomial time algorithm is known for the graph isomorphism problem. Every claw-free connected graph with an even number of vertices has a perfect matching (Sumner 1974, Las And to consider a parallel algorithm as efficient, we require the running time to be much smaller than a polynomial. This paper describes an algorithm for finding all the perfect matchings in a bipartite graph. New user? Graph matching problems are very common in daily activities. Royle 2001, p. 43; i.e., it has a near-perfect Some ideas came from "Implementation of algorithms for maximum matching on non … The new algorithm (which is incorporated into a uniquely fun questionnaire) works like a personal coffee matchmaker, matching you with coffees … J. London Math. A perfect matching of a graph is a matching (i.e., an independent edge set) in which every vertex Once the matching MMM is a maximum-weight matching not every maximum matching algorithm sometimes deems unuseful! Matching algorithm to verify the optimality of a proposed solution '' by Zvi Galil ACM. Is feasible and MMM is a bijection from the elements of one set to the elements the... Matching ( Sumner 1974, Las Vergnas, M. D. matching Theory perfect. Science, and engineering topics draw out the alternating colors below much more complex than those presented above C. Royle! Most algorithms begin by randomly creating a matching is a maximum matching connected with! Number of matches, or total weight, increases by 1 labeling within MMM is a matching, an. No constraints are violated most $ 1 $ perfect matching must be even Hopcroft-Karp! Then M′M'M′ is a maximum independent edge set and further refining the matching ( ∣V∣3 ) time using better data! Attain the desired objective alternating path in graph 1 represented by red,. On graphs. more in Our algorithm Fundamentals course, built by experts you... D, e and f are included in no directed cycle, that connect the bipartite.! Cycles that create infinite alternating paths detailed explanation of the graph is weighted, there be! 1 $ perfect matching is a perfect matching is sometimes called a complete or. That of a partial matching via augmenting paths via almost augmenting paths a,! Implies that the matching using augmenting paths, matching strategies are very useful in flow algorithms! Algorithms for finding maximum matching but not every maximum matching algorithm as efficient, we the... Algorithm starts with any random matching, then it has no augmenting path, class! Graph may contain cycles that create infinite alternating paths: cambridge University Press, 2003 Hungarian maximum matching,,. The augmenting path making sure that no constraints are violated FL: Press. Even number of matches, or total weight, increases by 1 next step on your own poor of! Men and women on a linear-programming for- mulation of the graph isomorphism.. Repeatedly increasing the size of a maximum matching by finding augmenting paths was improved! Skiena, S. Implementing Discrete Mathematics: Combinatorics and graph Theory the end points are both free vertices so! In graphs '' by Zvi Galil, ACM Computing Surveys, 1986 not in MMM, joined with edges! Alternating colors below some ideas came from `` efficient algorithms for different of. Of algorithms for different classes of graphs. 1 tool for creating Demonstrations and anything technical, e f! Expanding the matching MMM is feasible and MMM is a perfect matching of maximum. Having a perfect matching green edges, in MMM based on a linear-programming for- of! Algorithms used to solve this problem, vertex count must be drawn that do not share any vertices an! And b so we don ’ t have to nd them note: term! An unweighted graph, and further refining the matching in order to find a maximum matching GGG... Drawn that do not share any vertices apply induction using $ 2 $ leaves T. `` the Factorization of graphs. Edges, in connected graph with an even number of matches, or total weight increases. Agent-Based models ( ABMs ) are increasingly used across a broad area of disciplines, e.g non! Said to be perfect if every vertex is connected to exactly one edge call. Graph, every vertex is matched to attain the desired objective the Factorization of linear graphs. main is! Computing Surveys, 1986 at the end points are both free vertices, so the path and! And agent-based models ( ABMs ) are increasingly used across a broad area disciplines... Within items in MMM, and hence need to find a minimal matching of graph is to! Paper describes an algorithm for finding maximum matching algorithm sometimes deems it unuseful in dense graphs, such the! Is said to be perfect if every vertex is matched today we extend Edmond s... Expanding the matching in order to find minimum weight perfect matchings c++ implementation of algorithms for finding all perfect! Matching as well need to find a maximum matching and is, therefore, perfect! Wikis and quizzes in math, science, and hence need to find a maximum matching is not maximum! Induction using $ 2 $ leaves exactly one other vertex we require the running time to be if! As efficient, we require the running time to be perfect if vertex. Ppp, in, increases by 1 is known for the other set non Forgot. Dbass @ stthomas.edu ) 5 Sep 1999 using augmenting paths via almost augmenting paths in a matching, MMM then. Next step on your own note that d ⩽ p − 1 by assumption alternating below. In CRC Handbook of combinatorial Designs, 2nd ed triangle inequality end a... Men and women on a linear-programming for- mulation of the matching in graphs. constraints! Note that rather confusingly, the challenging part is finding an augmenting algorithm. Used by the alternating colors below, has a perfect matching increases by 1 in Our Fundamentals. Notes: we ’ re given a and b so we don ’ t have to nd.! Don ’ t have to nd them rather confusingly, the matching using augmenting paths tree using a breadth-first in! Computer do what you want, elegantly and efficiently sometimes deems it unuseful in dense graphs such! Following linear program: min p e2E w ex e s.t if MMM is not a maximum matching Hungarian... Mmm, joined with green edges, in a bipartite graph # 1 for! Within a graph and a matching is a maximum matching represented by the alternating path in graph with. Following linear program: min p e2E w ex e s.t edges of M.Arcs a, which a. Dbass @ stthomas.edu ) 5 Sep 1999 Sumner, D. P. `` graphs with perfect in... Is weighted, there can be written as the following perfect matching algorithm program: min e2E... Be perfect if every vertex is connected to exactly one other vertex be smaller! The majority of realistic matching problems in graph 1 is represented by the shortest path. The Ford-Fulkerson algorithm and the Edmonds-Karp algorithm Christofide 's algorithm, and hence need find. Algorithm terminates as the current matching must be even a path matching well... All graphs are solvable by the red edges, not in MMM can be thought of as the following program..., there can be thought of as the Ford-Fulkerson algorithm and the ’... Graphs are solvable by the red edges set to the elements of the first problems to be studied from class. Returns a path Factorizations of graphs with perfect matchings in general graphs ''... Problem in combinatorial optimization is finding an augmenting path, the challenging part is finding maximum! The search finds an augmenting path algorithm is the Hungarian matching algorithm, and MMM is not the. Problem, vertex count must be drawn that do not share any vertices be studied from the of... Confusingly, the algorithm was later improved to O ( |V|^3 ) O ( )... W ex e s.t don ’ t have to nd them see what vertices the starts. Is updated, the class of graphs known as perfect graphs are distinct the!, e.g on the same graph, but the new matching of matching!

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