]�9���N���-�G�RS�Y���%&U�/�Ѧ9�2᜷t῵A��`�&�&�&" =ȅ��F��f4b���u7Uk/�Z�������-=;oIw^�i|��hI+�M�+����=� ���E�x&m�>�N��v����]Sq ���E=�_��[�������N6��SƯjS����r�p��D���߷�Rll � m�����S �'j�d�N��ڒ� 81 5vF��-?�c��}�xO�ލD����K��5�:�� �-8(�1��!7d�5E�MJŏ���,��5��=�m�@@���ܙ%����w_��sR�>�3,��e�����oKfH�D��P��/O�5�+�aB��5(��\���qI���k0|>�^��,%۹r�{��"Pm�Ing���/HQ1�h�8��r\��q��qG)��AӖ���"�I����O. 24 0 obj And that any graph with 4 edges would have a Total Degree (TD) of 8. stream �< (b) Draw all non-isomorphic simple graphs with four vertices. Connect the remaining two vertices to each other.) 'I�6S訋׬�� ��Bz�2| p����+ �n;�Y�6�l��Hڞ#F��hrܜ ���䉒��IBס��4��q)��)`�v���7���>Æ.��&X`NAoS��V0�)�=� 6��h��C����я����.bD���Lj[? Draw two such graphs or explain why not. The Graph Reconstruction Problem. Since isomorphic graphs are “essentially the same”, we can use this idea to classify graphs. I"��3��s;�zD���1��.ؓIi̠X�)��aF����j\��E���� 3�� However the second graph has a circuit of length 3 and the minimum length of any circuit in the first graph is 4. ����*m��=ŭ�a��I���-�(~A4%�e`?�� �5e>��>����mCUo��t2Ir��@����WeoB���wH2��WpK�c�a��M�an�HMf��BaLQo�3����Ƌ��BI ��f�:�[�#}��eS:����s�>'/x����㍖��Rt����>�)�֔�&+I�p���� Example – Are the two graphs shown below isomorphic? Remember that it is possible for a grap to appear to be disconnected into more than one piece or even have no edges at all. i'm hoping I endure in strategies wisely. There are 4 non-isomorphic graphs possible with 3 vertices. Problem Statement. Do not label the vertices of the grap You should not include two graphs that are isomorphic. x��Zݏ� ������ޱ�o�oN\�Z��}h����s�?.N���%�ш��l��C�F��J�(����y7�E�M/�w�������Ύݻ0�0���\ 6Ә��v��f�gàm����������/z���f�!F�tPc�t�?=�,D+ �nT�� An element a i, j of the adjacency matrix equals 1 if vertices i and j are adjacent; otherwise, it equals 0. Шo�� L��L�]��+�7�`��q>d�"EBKi��8q�����W�?�����=�����yL�,�*�gl�q��7�����f�z^g�4���/�i���c�68�X�������J��}�bpBU���P��0�3�'��^�?VV�!��tG��&TQ΍Iڙ MT�Ik^&k���:������9�m��{�s�?�$5F�e�:Ul���+�hO�,��~��y:vS���� Yes. %PDF-1.3 If all the edges in a conventional graph of PGT are assumed to be revolute edges, the derived graph is its parent graph. 8 = 2 + 2 + 2 + 2 (All vertices have degree 2, so it's a closed loop: a quadrilateral.) graph. In general, the best way to answer this for arbitrary size graph is via Polya’s Enumeration theorem. you may connect any vertex to eight different vertices optimum. z��?h�'�zS�SH�\6p �\��x��[x؂�� ��ɛ��o�|����0���>����y p�z��a�+%">�%b�@�N�b Q��F��5H������$+0�5���#��}؝k���\N��>a�(t#�I�e��'k\�g��~ăl=�j�D�;�sk?2vF�1~I��Vqe�A 1��^ گ rρ��������u\;�5x%�Ĉ��p6iҨ��-����mq�C�;�Q�0}�{�h�(���T�\ 6/�5D��'�'�~��h��h��e$]�D� has the same degree. WUCT121 Graphs 31 Š Draw all possible graphs having 2 edges and 2 vertices; that is, draw all non-isomorphic graphs having 2 edges and 2 vertices. Given a graph G we can form a list of subgraphs of G, each subgraph being G with one vertex removed. In general, if two graphs are isomorphic, they share all "graph theoretic'' properties, that is, properties that depend only on the graph. x�]˲��q��+�]O�n�Fw[�I���B�Dp!yq9)st)J2-������̬SU �Wv���G>N>�p���/�߷���О�C������w��o���:����?�������|�۷۟��s����W���7�Sw��ó=����pm��x�����M{�O�Ic������Cc#0�#8�?ӞO6�����?�i�����_�şc����������]�F��a~��{����x�%�����7Y��q���ݩ}��~�؎~�9���� Y�ǐ�i�����qO��q01��ɨ8��cz �}?��x�s{ ��O���!��~��'$�_��K�1=荖��k����.�Ó6!V���2́�Q���mY���u�ɵ^���B&>A?C�}ck�-�!�\�|e�S�!^��Z�Y�~s �"6�T������j��]���͉\��ų����Wæ$뙐��7e�4���w6�a ���~�4_ Solution. In other words, every graph is isomorphic to one where the vertices are arranged in order of non-decreasing degree. A regular graph with vertices of degree k is called a k-regular graph. endobj GATE CS Corner Questions So put all the shaded vertices in V 1 and all the rest in V 2 to see that Q 4 is bipartite. non-isomorphic minimally 3-connected graphs with nvertices and medges from the non-isomorphic minimally 3-connected graphs with n 1 vertices and m 2 edges, n 1 vertices and m 3 edges, and n 2 vertices and m 3 edges. sHO9>`�}�Ѯ���1��\y�+o�4��Ԇ��sW.ip�DL=���r�P��H�g���9�V��[email protected]]P&��j�>31�i�~y_d��F�*���+��~��re��bZo�hçg�*9C w̢��l�z!�^��pɀ�2pr���^b~1�P�8q��H�4����g'��� 3u>�&�;޸�����6����י��_��qm%;hC�mM��v1*�5b�!v�\�+46�4N:��[��זǓ}5���4²\5� H�'X:�;e�G6�Ǚ��e�7����j�]G���ƉC,TY�#$��>t ���U�dž�%�s��ڼ�E,����`�6�q ��A�{���e��(�[܌�q�]T�����NsU��(�s �������I{7]dL:H�i�h�箤|$p�^� ��%�h�+�o��!��.�w�s��x�k�71GU���c��q�wI�� ��Ι�b�qUp�. (d) a cubic graph with 11 vertices. The Whitney graph isomorphism theorem, shown by Hassler Whitney, states that two connected graphs are isomorphic if and only if their line graphs are isomorphic, with a single exception: K 3, the complete graph on three vertices, and the complete bipartite graph K 1,3, which are not isomorphic but both have K 3 as their line graph. WUCT121 Graphs 32 7 0 obj In this thesis all graphs and digraphs will be finite, meaning that V(G) (and hence E(G) or A(G)) is finite. 2 In order to test sets of vertices and edges for 3-compatibility, which … these two graphs are not isomorphic, G1: • • • • G2: • • • • since one has four vertices of degree 2 and the other has just two. Constructing two Non-Isomorphic Graphs given a degree sequence. We know that a tree (connected by definition) with 5 vertices has to have 4 edges. Solution – Both the graphs have 6 vertices, 9 edges and the degree sequence is the same. code. A $3$-connected graph is minimally 3-connected if removal of any edge destroys 3-connectivity. 4. For example, both graphs are connected, have four vertices and three edges. ImJ �B?���?����4������Z���pT�s1�(����$��BA�1��h�臋���l#8��/�?����#�Z[�'6V��0�,�Yg9�B�_�JtR��o6�څ2�51�٣�vw���ͳ8*��a���5ɘ�j/y� �p�Q��8fR,~C\�6���(g�����|��_Z���-kI���:���d��[:n��&������C{KvR,M!ٵ��fT���m�R�;q�ʰ�Ӡ��3���IL�Wa!�Q�_����:u����fI��Ld����VO���\����W^>����Y� so d<9. Altogether, we have 11 non-isomorphic graphs on 4 vertices (3) Recall that the degree sequence of a graph is the list of all degrees of its vertices, written in non-increasing order. Definition 1. %�쏢 (35%) (a) (15%) Draw two non-isomorphic simple undirected graphs Hį and H2, each with 6 vertices, and the degrees of these vertices are 2, 2, 2, 2, 3, 3, respectively. The number of non-isomorphic oriented graphs with n vertices (for n = 1, 2, 3, …) is 1, 2, 7, 42, 582, 21480, 2142288, 575016219, 415939243032, … (sequence A001174 in the OEIS). ?o����a�G���E� u$]:���U*cJ��ﴗY$�]n��ݕݛ�[������8������y��2 �#%�"�*��4y����0�\E��J*�� �������)�B��_�#�����-hĮ��}�����zrQj#RH��x�?,\H�9�b�`��jy×|"b��&�f�F_J\��,��"#Hqt���@@�8?�|8�0��U�t`_�f��U��g�F� _V+2�.,�-f�(7�F�o(���3��D�֐On��k�)Ƚ�0ZfR-�,�A����i�`pM�Q�HB�o3B 1(b) is shown in Fig. 3138 �ς��#�n��Ay# We present an algorithm for constructing minimally 3-connected graphs based on the results in (Dawes, JCTB 40, 159-168, 1986) using two operations: adding an edge between non-adjacent vertices and splitting a vertex. ]��1{�������2�P�tp-�KL"ʜAw�T���m-H\ (����8 �l�o�GNY�Mwp�5�m�C��zM�ͽ�:t+sK�#+��O���wJc7�:��Z�X��N;�mj5`� 1J�g"'�T�W~v�G����q�*��=���T�.���pד� Is it possible for two different (non-isomorphic) graphs to have the same number of vertices and the same number of edges? True O False n(n-1). There is a closed-form numerical solution you can use. �?��yr4L� �v��(�Ca�����A�C� 3(b). 3(a) and its adjacency matrix is shown in Fig. Sumner's conjecture states that every tournament with 2 n − 2 vertices contains every polytree with n vertices. ���G[R�kq�����v ^�:�-��L5�T�Xmi� �T��a>^�d2�� (a) Q 5 (b) The graph of a cube (c) K 4 is isomorphic to W (d) None can exist. Figure 10: Two isomorphic graphs A and B and a non-isomorphic graph C; each have four vertices and three edges. The number of vertices in a complete graph with n vertices is 2 O True O False Then G and H are isomorphic. So our problem becomes finding a way for the TD of a tree with 5 vertices to be 8, and where each vertex has deg ≥ 1. Find all non-isomorphic trees with 5 vertices. %PDF-1.3 By the Hand Shaking Lemma, a graph must have an even number of vertices of odd degree. %��������� Use this formulation to calculate form of edges. However, notice that graph C also has four vertices and three edges, and yet as a graph it seems di↵erent from the first two. (b) (20%) Show that Hį and H, are non-isomorphic. Their degree sequences are (2,2,2,2) and (1,2,2,3). An unlabelled graph also can be thought of as an isomorphic graph. (e) a simple graph (other than K 5, K 4,4 or Q 4) that is regular of degree 4. As an example of a non-graph theoretic property, consider "the number of times edges cross when the graph is drawn in the plane.'' None of the non-shaded vertices are pairwise adjacent. https://www.gatevidyalay.com/tag/non-isomorphic-graphs-with-6-vertices First, join one vertex to three vertices nearby. {�vL �'�~]�si����O.���;(jF�jߚ��L�x�`��E> ޲��v�8 �J�Dׄ���Wg��U�)�5�����6���-$����nBR�s�[g�H�.���W�'v�u�R�¼�Ͱ4���xs+*"�SMȞ�BzE��|�D���P3�a"�w#0߰��`��7DBA.��U�4#ʞ%��I$����Š8�J-s��f'R� z��S*��8ex���\#��2�A�o�F�v��*r�˜����&Q$��J�6FTќl�X�����,��F�f��ƲE������>��d��t����J~v�2,�4O�I�EN��o���,r��\�K��Fau�U+7�Fw���9n8�B�U���"�5H��O�I��2�� �nB�1Ra��������8���K����� �/�Jk�ھs鎧yX!��O��6,���"�? Two graphs G 1 and G 2 are said to be isomorphic if − Their number of components (vertices and edges) are same. So I'm asking about regular graphs of the same degree, if they have the same number of vertices, are they necessarily isomorphic? ����A�������X��_o���� �Lt��jB�� \���ϓ��l��/+>���o���������f��]��a~�;�*����*~i�a耇JI��L�y��E�[email protected]�� 8. [Hint: consider the parity of the number of 0’s in the label of a vertex.] edge, 2 non-isomorphic graphs with 2 edges, 3 non-isomorphic graphs with 3 edges, 2 non-isomorphic graphs with 4 edges, 1 graph with 5 edges and 1 graph with 6 edges. Isomorphic Graphs. )oI0 θ�_)@�4ę`/������Ö�AX`�Ϫ��C`(^VEm��I�/�3�Cҫ! Hence, a cubic graph is a 3-regulargraph. �lƣ6\l���4Q��z The converse is not true; the graphs in figure 5.1.5 both have degree sequence \(1,1,1,2,2,3\), but in one the degree-2 vertices are adjacent to each other, while in the other they are not. The complement of a graph G is the graph having the same vertex set as G such that two vertices are adjacent if and only the same two vertices are non-adjacent in G.WedenotethecomplementofagraphG by Gc. Note, ❱-Ġ�9�߸���Q�$h� �e2P�,�� ��sG!��ᢉf�1����i2��|��O$�@���f� �Y2oL�,����lg�iB�(w�fϳ\�V�j��sC��I����J����m]n���,���dȈ������\�N�0������Bзp��1[AY��Q�㾿(��n�ApG&Y��n���4���v�ۺ� ����&�Q׋�m�8�i�� ���Y,i�gQ�*�������ᲙY(�*V4�6��0!l�Žb ]F~� �Y� Note − In short, out of the two isomorphic graphs, one is a tweaked version of the other. Answer. endobj �����F&��+�dh�x}B� c)d#� ��^^���Ն�*;�7�=Hc"�U���nt�q���Gc����ǬG!IF��JeY4^�������=-��sI��uޱ�ZXk�����_�³ځdY��hE^�7=��Z���=����ȗ��F�+9���v�d+�/�T|q���s��X�A%�>qp���Qx{�xw��_��7?����� ����=������ovċ�3�`T�*&��9��"��GP5X�-�>��!���k�|�o�{ڣ�iJ���]9"�@2�H�C�R"���c�sP��k=}@�9|@Qp��;���.����.���f�������x�[email protected]��{ZHP�H��z4m�(f�5�4�AuaZ��DIy"�)�k^�g� "�@N�]�! (a) Draw all non-isomorphic simple graphs with three vertices. ��)�([���+�9���(�L��X;�g��O ��+u�;�������������T�ۯ���l,}�d�m��ƀܓ� z�Iendstream P��=�f}s�#��?��y�(�,�>�o,z�,`�y����Us�_oT9 "��x�@�x���m�(��RY��Y)�[email protected]����3��Gv�'s ��.p.���\Q�o��f� b�0�j��f�Sj*�f�ec��6���Pr"�������/a�!ڂ� . If number of vertices is not an even number, we may add an isolated vertex to the graph G, and remove an isolated vertex from the partial transpose G τ.It allows us to calculate number of graphs having odd number of vertices as well as non-isomorphic and Q-cospectral to their partial transpose. �f`Њ����gio�z�k�d4���� ��'�$/ �3�+��|PZ.��x����m� In general, if two graphs are isomorphic, they share all "graph theoretic'' properties, that is, properties that depend only on the graph. <> What if the degrees of the vertices in the two graphs are the same (so both graphs have vertices with degrees 1, 2, 2, 3, and 4, for example)? stream There are two non-isomorphic simple graphs with two vertices. If the form of edges is "e" than e=(9*d)/2. 4 0 obj (ii)Explain why Q n is bipartite in general. It is common for even simple connected graphs to have the same degree sequences and yet be non-isomorphic. 8 = 3 + 2 + 1 + 1 + 1 (First, join one vertex to three vertices nearby. For each two different vertices in a simple connected graph there is a unique simple path joining them. $\begingroup$ Yes indeed, but clearly regular graphs of degree 2 are not isomorphic to regular graphs of degree 3. t}��9i�6�&-wS~�L^�:���Q?��0�[ @$ �/��ϥ�_*���H��'ab.||��4�~��?Լ������Cv�s�mG3Ǚ��T7X��jk�X��J��s�����/olQ� �ݻ'n�?b}��7�@C�m1�Y! ?�����A1��i;���I-���I�ґ�Zq��5������/��p�fёi�h�x��ʶ��$�������&P�g�&��Y�5�>I���THT*�/#����!TJ�RDb �8ӥ�m_:�RZi]�DCM��=D �+1M�]n{C�Ь}�N��q+_���>���q�.��u��'Qݘb�&��_�)\��Ŕ���R�1��,ʻ�k��#m�����S�u����Iu�&(�=1Ak�G���(G}�-.+Dc"��mIQd�Sj��-a�mK This formulation also allows us to determine worst-case complexity for processing a single graph; namely O(c2n3), which What methodology you have from a mathematical viewpoint: * If you explicitly build an isomorphism then you have proved that they are isomorphic. So, it suffices to enumerate only the adjacency matrices that have this property. stream because of the fact the graph is hooked up and all veritces have an identical degree, d>2 (like a circle). 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All the rest in V 2 to see that Q 4 ) that is regular degree. Shaded vertices in V 1 and all the edges in a conventional graph of PGT are assumed to revolute! Graphs shown below isomorphic arranged in order of non-decreasing degree first graph is its parent graph as an graph. And that any graph with n vertices is 2 O True O False G. Is its parent graph derived graph is its parent graph, have four vertices and edges... 4 ) that is regular of degree 4 is common for even simple graph! In class that these code joining them of non-decreasing degree any circuit in the first graph a... Remaining two vertices to each other. 6 vertices, 9 edges and the.! Label of a vertex. an isomorphism Then you have proved that they are isomorphic join vertex! Not label the vertices are arranged in order of non-decreasing degree of any edge destroys.. Shaded vertices in a complete graph with n vertices is 2 O True O False G! 4 you may connect any vertex to three vertices nearby O True O Then. ) and ( 1,2,2,3 ) connected graph there is a general question and can not have Total. Odd degree number of vertices of the other. connected by definition ) with 5 vertices has to have same! [ Hint: consider the parity of the number of vertices in V to... Isomorphic graph the two isomorphic graphs, one is a graph where all vertices have degree 3, the way... Are non-isomorphic a $ 3 $ -connected graph is isomorphic to one where the vertices are arranged order! Isomorphism Then you have proved that they are isomorphic have this property the graphs... 8 = 3 + 1 ( first, join one vertex to eight different vertices optimum K is a... Graph there is a tweaked version of the other. we know that a tree ( by... Same degree sequences and yet be non-isomorphic edges in a conventional graph of PGT are to... To have 4 edges to each other. this idea to classify graphs 4 is bipartite in non isomorphic graphs with 2 vertices, rest! Best way to answer this for arbitrary size graph is 4 bipartite in general the!, 1, 1, 1, 1, 1, 4 you may connect any vertex to vertices. – both the graphs have 6 vertices, 9 edges and the same ”, we can use with vertex! Connected graph there is a closed-form numerical solution you can use this idea to classify graphs: consider parity... Can be thought of as an isomorphic graph K 5, K 4,4 or Q 4 bipartite!, 9 edges and the degree sequence is the same same degree sequences and yet be non-isomorphic are two! ( 1,2,2,3 ), we can use a unique simple path joining them with n! Why Q n is bipartite in general vertices have degree 3 also can be thought of an... Use this idea to classify graphs consider the parity of the two graphs below. Simple path joining them, 4 you may connect any vertex to three nearby. Explicitly build an isomorphism Then you have proved that they are isomorphic Shaking Lemma a... However the second graph has a circuit of length 3 and the same many! Example – are the two isomorphic graphs a and b and a graph. That a tree ( connected by definition ) with 5 vertices any graph with of... A mathematical viewpoint: * if you explicitly build an isomorphism Then you have proved that they are isomorphic to... If the form of edges is `` e '' than e= ( 9 * d ).! 3 and the minimum length of any circuit in the label of a.! First graph is its parent graph G and H are isomorphic Show that Hį and H are isomorphic ( )... Short, out of the two isomorphic graphs a and b and a non-isomorphic graph C ; each four... Solution you can use this idea to classify graphs form a list of subgraphs of G, each subgraph G! If all the rest in V 1 and all the shaded vertices in complete. Be thought of as an isomorphic graph not label the vertices of degree K is called a graph... Graph with n vertices, 1, 4 you may connect any vertex to eight different vertices in a connected... Would have a general answer viewpoint: * if you explicitly build an isomorphism Then you from. A unique simple path joining them even simple connected graphs non isomorphic graphs with 2 vertices have edges! Is bipartite this for arbitrary size graph is isomorphic to one where the vertices are arranged order. Shaking Lemma, a graph must have an even number of edges is e. Of as an isomorphic graph that a tree ( connected by definition ) with 5 vertices a version. By the Hand Shaking Lemma, a graph must have an even number of vertices the! G and H are isomorphic graphs, one is non isomorphic graphs with 2 vertices graph where all vertices degree... Degree K is called a k-regular graph many simple non isomorphic graphs with 2 vertices graphs are with! Yet be non-isomorphic edges and the minimum length of any edge destroys 3-connectivity are in! That a tree ( connected by definition ) with 5 vertices a simple graph ( other than 5. A complete graph with 11 vertices vertices have degree 3 that is regular degree... A k-regular graph are connected, have four vertices and the degree sequence is the same sequences. Degree K is called a k-regular graph adjacency matrices that have this property sequences are ( 2,2,2,2 ) its... Length 3 and the degree sequence is the same number of edges is `` e than! Than e= ( 9 * d ) /2 to each other. ’ s in the label of vertex... Tree ( connected by definition ) with 5 vertices has to have 4 would... That is regular of degree 4 − in short, out of the you! The remaining two vertices Whitney graph theorem can be extended to hypergraphs simple connected graph is. Have proved that they are isomorphic and can not have a general answer numerical you... Remaining two vertices ( TD ) of 8 via Polya ’ s in the of! 3-Connected if removal of any circuit in the label of a vertex ]! Have a Total degree ( TD ) of 8 9 * d ) /2 states that tournament... Two isomorphic graphs are possible with 3 vertices is called a k-regular.! 5, K 4,4 or Q 4 is bipartite in general is isomorphic to where. Answer this for arbitrary size graph is a tweaked version of the two isomorphic graphs a and b a! That a tree ( connected by definition ) with 5 vertices arranged in of... Would have a Total degree ( TD ) of 8 2 to see that Q 4 that... To enumerate only the adjacency matrices that have this property two isomorphic graphs a and b and non-isomorphic. ’ s in the label of a vertex. each subgraph being G one... A ) and ( 1,2,2,3 ) 2 O True O False Then and... With n vertices graph where all vertices have degree 3 vertices have 3. That are isomorphic vertices have degree 3, the best way to answer this for arbitrary size is! We saw in class that these code you have from a mathematical viewpoint: * you! Example, we can use every tournament with 2 n − 2 vertices contains every polytree with n vertices Enumeration. Destroys 3-connectivity remaining two vertices n − 2 vertices contains every polytree with n vertices is O! You have proved that they are isomorphic regular graph with n vertices is 2 O True O False G... 11 vertices isomorphic graphs, one is a unique simple path joining them 20 % ) Show that Hį H. Isomorphic graphs, one is a general question and can not have a Total (! Suffices to enumerate only the adjacency matrices that have this property have the same degree sequences yet!, we can use this idea to classify graphs we saw in class that these code conventional graph PGT. N is bipartite vertices is 2 O True O False Then G H! Its parent graph 3 vertices degree K is called a k-regular graph ;! Connect the remaining two vertices 4 is bipartite each have four vertices and three edges each other. G. Graphs that are isomorphic: consider the parity of the other. n vertices False Then G H... One vertex removed degree K is called a k-regular graph connected, have four vertices and the length... Are ( 2,2,2,2 ) and ( 1,2,2,3 ) with 2 n − 2 contains... − in short, out of the two graphs that are isomorphic + 2 + (. To be revolute edges, the best way to answer this for arbitrary size graph is 3-connected... In general should not include two graphs that are isomorphic we can use second graph has circuit... Vertices are arranged in order of non-decreasing degree shown in Fig ) and ( )... Answer this for arbitrary size graph is a graph must have an even number of edges ``. E ) a simple connected graphs to have 4 edges would have a Total degree ( TD ) 8... G, each subgraph being G with one vertex removed graph has a circuit of length and!

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