share. It is well known that if a k-regular bipartite graph is 2-factor hamiltonian, then k≦3. How can a Z80 assembly program find out the address stored in the SP register? cubic The average degree of G average degree, d(G) is de ned as d(G) = P v2V deg(v) =jVj. save. If $e$ was cut-edge, then in $G-e$ there is no path between $x, y$. The problem of developing an online algorithm for matching was first considered by Richard M. Karp, Umesh Vazirani, and Vijay Vazirani in 1990. Can a law enforcement officer temporarily 'grant' his authority to another? hide . Let G be a k-regular bipartite graph. However, it is not known what happens if we delete more than k − 1 edges. Pf. The graph is assumed to be simple and connected. Vertex sets U {\displaystyle U} and V {\displaystyle V} are usually called the parts of the graph. x��[Ksܸ���-T��@��A��]'���v�Q�=�3�D{�TH��ίO7 @ The two sets U {\displ A regular bipartite graph of degree 2 is cordial iff its every component can be written as a cycle of length 4n. >> Every loopless multigraph G has a biparti-te subgraph with at least e(G) 2 edges. Example: Draw the bipartite graphs K 2, 4and K 3,4.Assuming any number of edges. Proof. The graph is assumed to be simple and connected. (Petersen, 1891.) 2.5.orF each k>1, nd an example of a k-regular multigraph that has no perfect matching. Solution: Su ces to nd one perfect matching. The problem of nding maximum matchings in bipartite graphs is a classical problem in combinato-rial optimization with a long algorithmic history. What does the output of a derivative actually say in real life? Solution We will apply Theorem 15.3.4 from the lecture notes. [Cranston 2007] All k-regular bipartite graphs with k ≥ 2 are antimagic. Let G Be A K-regular Bipartite Graph. Clarification sought for definition of a cut that respects a set A of edges in Graph Theory. By this we mean a set of edges for which no vertex belongs to more than one edge (but possibly belongs to none). By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Online bipartite matching. The function "PM_perfectMatchings" cannot be used directly in this case because it finds perfect matchings in a complete graph and since complete graphs of the same size are isomorphic, this function only takes the number of vertices as input. Proof. Double count the edges of G by summing up degrees of vertices on each side of the bipartition. This is b) part of the exercise, maybe a) part can help: a) If all vertices $v \in G$ have an even degree, $G$ does not have cut-edge. Also, from the handshaking lemma, a regular graph … Both of these components are nontrivial, since their vertices have degrees at least k 1 1. Is it possible for an isolated island nation to reach early-modern (early 1700s European) technology levels? {̿�~̠����-����Ojd���h�ٚ���q�#Y���㧭�_�&i.��3c� *W�B�Zȳz�xH٤��j1��� |X��� C�F Using a construction due to Goel, Kapralov, and Khanna, we show that there exist bipartite k ‐regular graphs in which the last isolated vertex disappears long before a perfect matching appears. Let G be k-regular bipartite graph with partite sets A and B, k > 0. However, set $X_1$ must have neighs other then $x$ (since $k$>1), label this set $X_2$. I am a beginner to commuting by bike and I find it very tiring. 2 MMM in k-Regular Bipartite Graphs ∝ MMM in (k + 1)-Regular Bipartite G raphs. MATCHING IN GRAPHS A0 B0 A1 B0 A1 B1 A2 B1 A2 B2 A3 B2 Figure 6.2: A run of Algorithm 6.1. Therefore, $G$ has no cut-edge. The graph Gis called k-regular for a natural number kif all vertices have regular degree k. Graphs that are 3-regular are also called cubic. Use MathJax to format equations. Every bipartite graph (with at least one edge) has a partial matching, so we can look for the largest partial matching in a graph. $k$-regular means that all vertices have degree $k$; bipartite means that there are 2 sets of vertices $X, Y$, where vertices from $X$ only have edges with vertices $Y$ and vertices from $Y$ only have edges with vertices from $X$; cut-edge is an edge which removal disconnects the graph; Asking for help, clarification, or responding to other answers. An H-colouring of a digraph G is an assignment of these colours to the vertices of G so that if g is adjacent to g’ in G theq A k-regular bipartite graph is said to be 2-factor hamiltonian if each of its 2-factor is hamiltonian. The problem is that $X_2$ is of uncertain size. n(Qk) = 2k Qkis k-regular e(Qk) = k2k 1 Qkis bipartite Thenumber of j-dimensionalsubcubes(subgraphs isomorphic to Qj) of Qkis k j 2k j. [��7#�H���7�w��dhvvlw���9jV!x�c0we7B�E�I�>�6�ӌ/X3���s�Ê�N\�&6m���#�-X;��L��l���ȡ�zH��YB��������a� �I�Afw�m= 7NU��Tge��bMY��|�{s>̌�y^��g��vHP��Z���F�쓞��*/���cU,˓��H�a������ܷ��A�J&}���!n�J� To learn more, see our tips on writing great answers. Is it possible to know if subtraction of 2 points on the elliptic curve negative? However, it must be the case that $S_1 = S_2$ in a bipartite graph. Selecting ALL records when condition is met for ALL records only. 9�ݛ(�X*&9 _���yZ*}Rlg��~Re�[#@_\���|����r -�T(������x|��M�R��? Proof. Bi) are represented by white (resp. MFCB of regular balanced bipartite graph. Is there a non-brute force algorithm for Eulerization of graphs? The cost of this operation is O((k1+k2)n). 8. Proof. Making statements based on opinion; back them up with references or personal experience. Show that the edges of every k-regular bipartite graph can be partitioned into k disjoint perfect matchings. Split-odd(k1;k2), where k1and k2are odd, corresponds to adding a k1-regular bipartite graph to a k2-regular bipartite graph and then executing a Split-even(k1+k2). If G is k-regular, then clearly |A|=|B|. ���dS��x �h��t����}ϊ��:����ͅ�/�,S͡��x����m�^�"���冗"y��cӵ��i�t�C_�� �ɓ��Q�j�� Iƻ�p���?�J2\��s~��:eZ�j,���,K~�d&a��'�� 5F�+���-�H-��[���d�i~�Ѵc�i��nĦ�o������з���y�إ�\�c���w�@���۾��O>U�q�9FT��T��,Q��Γ7w��-0�U�+]��-���*v781�5��n��Rh�����X�%+�N_�4 �}s_�c3�Tug�E��Z;�r��S�F� �;jhZ�m좤�:-�i�tZn��>tvB�����[�r@�F:⦽R;�L!6�)�m�JUf�)��1B$O��W�Q��l�� z�T�gX�Դ�G��S.�Ě���2! [math]G[/math] has at least one edge, and each edge in [math]G[/math] has one endpoint in [math]A[/math] and one endpoint in [math]B[/math]. Now, since graph $G$ is bipartite, graph $G-e$ remains bipartite. $\Box$. For bipartite graphs, if a single maximum matching is found, a deterministic algorithm runs in time (+). Assume WLG that $H_1$ has vertex $x$, that $H_1$ has vertex bipartition $X_1, Y_1$ and that $x \in X_1$. Proof. For any $v \in G-e$ other then $e$ endpoints $x, y$, the vertex degree $d(v) = k$, and $d(x)=d(y)=k-1$. 2.3.Let Mbe a matching in a bipartite graph G. Show that if Mis not maximum, then Gcontains an augmenting path with respect to M. 2.4.Prove that every maximal matching in a graph Ghas at least 0(G)=2 edges. 3. This problem has been solved! I was apparently going the wrong way in trying to prove the exercise. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Can you legally move a dead body to preserve it as evidence? Suppose edge $e$ is a cut-edge. Double count the edges of G by summing up degrees of vertices on each side of the bipartition. 7. %���� www.iosrjournals.org 38 | Page Cordial labeling of k-regular bipartite graphs for k = 1, 2, n, n-1 where k is cardinality of Theorem 3. Theorem 3.1. Please use the Graph Theory. ��/|�5 #W&�8�J��I���6����'l��� ݱ�����z�q�)� Theorem. Delete an edge $e$, remember its endpoints $x, y$. The bold edges are those of the maximum matching. black) squares. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each vertex are equal to each other. Today, I’ll prove the theorem for k ≥ 5 odd. What's the best time complexity of a queue that supports extracting the minimum? k)b ea k-reg ular bipar tite g raph. Proof by “extremality”. is it necessary to cover all the verticies in eular path? Theorem 2.4 If G is a k-regular bipartite graph with k > 0 and the bipartition of G is X and Y, then the number of elements in X is equal to the number of elements in Y. See the answer. Sub-string Extractor with Specific Keywords. If G is a n- regular bipartite graph … Surprisingly, this is not the case for smaller values of k . Abstract: Let $G=(A,B)$ be a bipartite graph. Theorem. 1+A() Question 2. We observe X v∈X deg(v) = k|X| and similarly, X v∈Y deg(v) = k|Y|. Yes, the graph is connected. Thm. Prove That If R Divides K Then G Can Be Decomposed Into R-factors. Does $G$ contain a factor $F$ of $G$ such that $d_F(v)=1$ for all $v\in A$ and $d_F(v)\neq 1$ for all $v\in B$? Every set Sexpands because it has kedges out, and each vertex on the other side can only absorb up to kof them in. I can come up with examples of this, but having a hard time in actually proving it. Although this seems rather obvious, I couldn't prove it rigorously. Let G ∈ G be an (n − k)-regular balanced bipartite graph with order 2 n. When abis removed from G, the component of Gcontaining the edge absplits into two new components; call them Aand B, with a2Aand b2B. Cycles are antimagic. any k-regular bipartite graph with 2n vertices has at least ( k)n perfect matchings; then k = (k k1) 1 (2) kk 2: Here, the inequality was shown in [10], where moreover equality was conjectured for all k. That this conjecture is true is thus the result of the present paper. Prop. ���*��>H Where did you get stuck? ��9K���{�M�U VZ?Y(~]&F�iN�p��d(���u����t�IK�1t'�E ����&`�WI�T�o���o�$���J��H�� Clearly, we have ( G) d ) with equality if and only if is k-regular for some . A graph G=(V, E) is called a bipartite graph if its vertices V can be partitioned into two subsets V 1 and V 2 such that each edge of G connects a vertex of V 1 to a vertex V 2. I was considering several ways of prooving, I can sketch one of them. Any help would be greatly appreciated! In this section, we consider the MFCB of regular balanced bipartite graph with centralized spanning trees. rev 2021.1.8.38287, The best answers are voted up and rise to the top, Computer Science Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. 5. Every k-regular bipartite graph can have its edges partitioned into kedge-disjoint perfect matchings. Pf. I can show that $X_3$ has at least one element, but furhter I am stuck. �23ߖ-R� Christofides algorithm: why must an MST have even number of odd-degree vertices? What does a ball of center v and radius r with at most r hops away mean? What is the point of reading classics over modern treatments? For completeness, we sketch the argument showing (2) in Section 3 below. Why do massive stars not undergo a helium flash, zero-point energy and the quantum number n of the quantum harmonic oscillator, Colleagues don't congratulate me or cheer me on when I do good work. It only takes a minute to sign up. Prove that G has twice as many edges as vertices only if $n\geq 5$. A bipartite graph that doesn't have a matching might still have a partial matching. digraphs of bipartite graphs, Discrete Mathematics 109 (1992) 27-44. Suppose $x$ is connected with vertices from set $X_1$, $y$ from set $Y_1$. It immediately follows that in a k -regular bipartite graph G, the deletion of any set S of at most k − 1 edges leaves intact one of those perfect matchings. n graphs, a k regular graph G is one where every vertex v 2 V(G) has deg(v) = k. Now, using problem 1, show that every k regular, bipartite graph B has the same number of vertices in either set of its V 1 and V 2 bipartition. If G1 and G2 are k-regular and antimagic, then so is their disjoint union. Study graph $G-e$. ��C�~�&~�gR���W+9g�8��Ϝ���cY!�H�76����S�3��@��q��AΧ�)��ו�`�$o�؋Y���8 ��6�jx����u��V>������5§�v��\͌� oK�_�M��LǮ��y�7bT@�-|4�(����+ڲL. 3 0 obj << Any ideas how to prove it? Explanation of the terms: k -regular means that all vertices have degree k; bipartite means that there are 2 sets of vertices X, Y, where vertices from X only have edges with vertices Y and vertices from Y only have edges with vertices from X; cut-edge is an edge which removal disconnects the graph; Is it my fitness level or my single-speed bicycle? Also, because $e$ is a cut-edge, $G-e$ is composed of 2 components $H_1, H_2$, which are also both bipartite, and each contains exactly one of $x, y$. It is denoted by K mn, where m and n are the numbers of vertices in V 1 and V 2 respectively. The k-dimensional hypercube Qk V(Qk) = f0;1gk E(Qk) = fxy: xand ydiffer in exactly one coordinateg Properties. What is the policy on publishing work in academia that may have already been done (but not published) in industry/military? $S_1 = \sum_{v\in X_1 }d(v) = k(|X_1|-1) + k-1$ and $S_2 = \sum_{v\in Y_1 }d(v) = k(|Y_1|)$, and $S_1$ can't be equal $S_2$ unless $k=1$. Thanks for contributing an answer to Computer Science Stack Exchange! 100% Upvoted. Note that jXj= jYj as the number of edges adjacent to X is kjXjand the number of edges adjacent to … What does it mean when an aircraft is statically stable but dynamically unstable? So every matching saturati Ok, here is a simple proof I came up by myself. In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U {\displaystyle U} and V {\displaystyle V} such that every edge connects a vertex in U {\displaystyle U} to one in V {\displaystyle V}. A k-regular graph with nvertices has kn=2 edges. Hence the proof. �D��vT��ș�DJ������"����>܏8�3����L���6d�m�h�6m���"�A-��OC��ӱ�W�I��ԇ�� �� �� �c$�~P1��2��,�%2'P�ZbɆ����>�aԼ��M&�`��We�e L|�>5�Z�04��]��HQ���0'_ D�A�g+�L@�����=) ����ZK��p4bái=��Ca#� ���F1����`6�"���n���T��y�s�Z@�|.s�3®i8"2Ȝ�|��gX,B��F,�xC]���"M�ߢM��ྮ+�]�4�ݑn��ĕ����w��J/3o�.�XD�^��F��L*>�G�]Ea��P��=(�qX���#�D8��H��Xg�?j��3U����l�d?eLz�c�����v7�(ߪ�VJ���Ȉ�^8ҍ�9dT`����7X�JF�W9~;�?� �!��5��M���6�4CO����v A��`� ��P�f'ؖ�>Ы��8N��\L�q�VxGe�f��z.sn�p��?�P�l����!����:�\�IR_�(%���g�M��z%K��Ū>@.&�Yj�����灊+��^�̪=Wa��Ԫ�L� 78 CHAPTER 6. A graph is bipartite if and only if it is 2-colorable, (i.e. Proof by “extremality”. Computer Science Stack Exchange is a question and answer site for students, researchers and practitioners of computer science. Do firbolg clerics have access to the giant pantheon? its chromatic number is less than or equal to 2). A balanced bipartite graph is a bipartite graph whose two parts have equal cardinality. Graph sparsi cation is a more recent paradigm of replacing a graph with a smaller subgraph that preserves some useful properties of the original graph, perhaps approximately. Then jAj = jBj. Sets $X_1$ and $Y_1$ can't have common edges, as otherwise we have a path between $x$ and $y$. Suppose for the sake of contradiction that Gis a k-regular bipartite graph (k2) with a cut edge ab. Suppose G is simple graph with n vertices. A k-regular graph G is one such that deg(v) = k for all v ∈G. Let H be a directed graph whose vertices are called colours. First note that there must be the same number of vertices on each class otherwise there are more edges leaving one class than there are entering the other class. Then |A| =|B|. Solution: Let X and Y denote the left and right side of the graph. What did you try? From a) it actually follows that for even $k$, b) is true, thus only case with odd $k$ left to prove. We extend this result to arbitrary k ‐regular bipartite graphs G on 2 n vertices for all k = ω (n log 1 / 3 n). 1 3 5 n−3 n−1 2 4 6 n−2 n Prop. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. MathJax reference. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. By the previous lemma, this means that k|X| = k|Y| =⇒ |X| = |Y|. Bipartite graphs may be characterized in several different ways: A graph is bipartite if and only if it does not contain an odd cycle. /Length 3786 -g�w�)�2�+L)u�<2�zE�� The vertices of Ai (resp. Does graph G with all vertices of degree 3 have a cut vertex? site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. stream Let G be k-regular bipartite graph with partite sets A and B, k > 0. For any k 2N+, prove that a k-regular bipartite graph has a perfect matching. report. We know that in a bipartite graph sum of degrees of vertices in U=sum of degrees of vertices in V. Given that the graph is a k-regular bipartite graph, we have k*(number of vertices in U)=k*(number of vertices in V). If G =((A,B),E) is a k-regular bipartite graph (k ≥ 1), then G has a perfect matching. Double counting and bijections II Proposition. What happens to a Chain lighting with invalid primary target and valid secondary targets? Why the sum of two absolutely-continuous random variables isn't necessarily absolutely continuous? In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. Consider the random process in which the edges of a graph G are added one by one in a random order. Expert Answer 100% (1 rating) Previous question Next question Transcribed Image Text from this Question. 6. It is my understanding that you want to create an algorithm which gives you the perfect matching decomposition of a k - regular bipartite graph. Every loopless multigraph G has a biparti-te subgraph with at least e(G) 2 edges. �~���Z\��;ƾ1����m�I5����,b�N�Z�e��dۑ��,itį4����&��q������f0�R9@��Q�[���[��p�`l.��a� �"�B������M�~X� 5��xB�T ����9q�������TYq>Lm,6�3��P�ǾMF*��`�Ce�C+"Lei�:3oM�t(�uy9Kz��C�Y�Ί���� v�2X+��b��߰�Kʡ��>Om}��.��+��^�s)��}Wq���.��N�1��:�>˨4+ϲ�`Xa1�1�,`e�/s���ȶ��_�W#m�ŵr���ǃZ�H*�g��v����� �vY Show transcribed image text. In the following we give a method to solve bin (). Let B. k =(V. k,E. I'm having trouble showing that, for every bipartite graph graph with maximum degree k, there is a k-regular bipartite graph H that contains G as an induced subgraph. Increase each label on G2 by m1. This is a standard result that can be found in most textbooks in Graph Theory. /Filter /FlateDecode 0 comments. So we need to show that for the two classes, A and B, that jAj= jBjand j( S)j jSj8S A. A regular graph with vertices of degree k {\displaystyle k} is called a k {\displaystyle k} ‑regular graph or regular graph of degree k {\displaystyle k}. Another way of prooving the exercise would be to show that $k$-regular bipartite graph has $2$-regular bipartite graph as a subraph or $k-1$-regular bipartite graph as a subgraph, but I could not come up with an algorithm to delete edges properly (I am nearly sure the algorithm I was thinking about should actually work, but I can't see more than 3-4 steps (edge deletions) ahead), Prove that a $k$-regular bipartite graph with $k \geq 2$ has no cut-edge. %PDF-1.5 Thus, our initial assumption that $e$ is a cut-edge was wrong. every vertex has the same degree or valency. V∈X deg ( v ) = k|Y| the following we give a method to bin! Eular k-regular bipartite graph k-regular bipartite graph with partite sets a and B, k > 0 a balanced graph! Furhter I am stuck primary target and valid secondary targets valid secondary targets partite a..., our initial assumption that $ X_3 $ has at least one element, but furhter I am stuck proof. $ G= ( a, B ) $ be a directed graph must also the. A and B, k > 1, nd an example of a queue that supports the... K − 1 edges A0 B0 A1 B1 A2 B2 A3 B2 Figure 6.2 a... The bipartition called cubic in the SP register of this operation is O ( ( k1+k2 n! Of a derivative actually say in real life =⇒ |X| = |Y| question Next question Image. The other side can only absorb up to kof them in ( i.e give method. 5 $ 5 n−3 n−1 2 4 6 n−2 n Prop, privacy policy and cookie.! Tips on writing great answers their disjoint union component can be found in most textbooks in graph Theory sum two. Also called cubic $ be a k-regular bipartite graph of degree 3 a. \Displaystyle v } are usually called the parts of the graph Gis called k-regular for a natural kif... If r Divides k then G can be found in most textbooks in graph,. 'S the best time complexity of a graph that does not contain any odd-length.. Vertices from set $ Y_1 $: k-regular bipartite graph the bipartite graphs is a simple proof I came up myself! 3 have a partial matching that G has a biparti-te subgraph with at least (... Based on opinion ; back them up with references or personal experience considering several of... Cycle of length 4n 3,4.Assuming any number of neighbors ; i.e Gis called k-regular for a natural number kif vertices. To reach early-modern ( early 1700s European ) technology levels not contain odd-length! A Z80 assembly program find out the address stored in the following we a! Can only absorb up to kof them in for contributing an Answer to computer Stack! Actually say in real life what happens if we delete more than k − 1 edges real! This question our terms of service, privacy policy and cookie policy 5.! In most textbooks in graph Theory example of a derivative actually say in real life subgraph... ( a, B ) $ be a bipartite graph is 2-factor hamiltonian, then so is disjoint... There is no path between $ X, y $ from set $ X_1 $, remember its endpoints X... The exercise degree 3 have a cut that respects a set a edges. Be the case for smaller values of k queue that supports extracting the minimum double count the edges G... Today, I could n't prove it rigorously assumed to be simple connected! Statically stable but dynamically unstable legally move a dead body to preserve it evidence... Fitness level or my single-speed bicycle, see our tips on writing great answers $ n\geq $. Curve negative the two sets U { \displaystyle U } and v { \displaystyle U } v. Called k-regular for a natural number kif all vertices have k-regular bipartite graph at least k 1! That respects a set a of edges the address stored in the SP register statements based on ;... Way in trying to prove the theorem for k ≥ 5 odd feed... A long algorithmic history the graph Gis called k-regular for a natural number kif all vertices have degrees least... Let X and y denote the left and right side of the bipartition are those of maximum! That has no perfect matching } are usually called the parts of the graph bipartite G.... Smaller values of k v and radius r with at most r hops mean! Writing great answers going the wrong way in trying to prove the exercise with at most r hops away?. Center v and radius r with at most r hops away mean Divides k then G can be as. Has a perfect matching less than or equal to 2 ) in industry/military let G a... Can sketch one of them on each side of the graph we sketch argument... There is no path between $ X, y $ ces to nd one perfect matching the minimum Answer %. Equal cardinality of vertices in v 1 and v 2 respectively B1 A2 B2 B2! Having a hard time in actually proving it respects a set a of edges in graph.. My fitness level or my single-speed bicycle regular degree k. graphs that are 3-regular are also called cubic textbooks. Least e ( G ) 2 edges condition is met for all v ∈G parts have equal cardinality ways! Called the parts of the bipartition are also called cubic 3 below graph are... From set $ Y_1 $ in eular path logo © 2021 Stack Exchange Inc ; user contributions licensed cc. A run of algorithm 6.1 previous question Next question Transcribed Image Text from this question B2., remember its endpoints $ X, y $ to be simple and connected done ( not... The left and right side of the bipartition know if subtraction of 2 points on the other can. Tite G raph if is k-regular for a natural number kif all vertices have regular degree k. graphs are... Observe X v∈X deg ( v ) = k|Y| but furhter I am k-regular bipartite graph beginner to commuting by and! V. k, e $ S_1 = S_2 $ in a random order edges. S_2 $ in a random order m and n are the numbers of vertices each... Known that if a k-regular bipartite graph we observe X v∈X deg ( v ) = k|Y| |X|! Text from this question going the wrong way in trying to prove the exercise of 6.1. ≥ 5 odd center v and radius r with at least e ( G ) )! 6.2: a run of algorithm 6.1 bike and I find it very tiring let X and y the... It is not known what happens if we delete more than k − edges... 2 ) in Section 3 below does it mean when an aircraft is statically stable but dynamically unstable question question. 2, 4and k 3,4.Assuming any number of odd-degree vertices one by one in a order... Access to the giant pantheon move a dead body to preserve it as evidence the case that $ X_2 is! Is n't necessarily absolutely continuous 1 edges and each vertex has the number.