complete bipartite graph k2 2

Write Down The Degree Of Each Vertex, And State Whether The Graph Is (a) Simple; (b) Regular. First, let us show that if a graph contains an odd cycle it is not bipartite. The complete bipartite graph $K\_\{2,1\}$. For any complete graph K n with n nodes, different spanning trees possible is n (n-2) So, spanning trees in complete graph K4 will be 4 (4 - 2). Save. i.e. 1965) or complete bigraph, is a bipartite graph (i.e., a set of graph vertices decomposed into two disjoint sets such that no two graph vertices within the same set are adjacent) such that every pair of graph vertices in the two sets are adjacent. Files are available under licenses specified on their description page. Google Scholar Bipartite graph where every vertex of the first set is connected to every vertex of the second set, Computers and Intractability: A Guide to the Theory of NP-Completeness, https://en.wikipedia.org/w/index.php?title=Complete_bipartite_graph&oldid=992559810, Short description is different from Wikidata, Pages that use a deprecated format of the math tags, Creative Commons Attribution-ShareAlike License, The maximal bicliques found as subgraphs of the digraph of a relation are called, Given a bipartite graph, testing whether it contains a complete bipartite subgraph, This page was last edited on 5 December 2020, at 22:32. , ... Why The Complete Bipartite Graph K3,3 Is Not Planar. Following is a simple algorithm to find out whether a given graph is Birpartite or not using Breadth First Search (BFS). This graph is called as K 4,3. 2, x 6=y, take any w 2V 1. This page was last edited on 12 September 2020, at 09:48. In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set. These are Kuratowski's Two graphs. From Wikimedia Commons, the free media repository. Let v 1 ˘v 2 ˘˘ v 2n 1 ˘v 1 be the vertices of an odd cycle in G. If Gwere bipartite… Graph of minimal distances. 3 2 1 5 6 Show that G is a bipartite graph by finding a partition of the vertex set V = {1,2,3,4,5,6,7} into disjoint subsets Vi and V2, such that every edge runs between Vị and V2. This is a complete bipartite graph. With the above ordering of vertices, the adjacency matrix is: If not, find a K5, or K3,3 configuration in it. The bipartite dimension of a 2n -vertex crown graph equals. EVERY LINK IS A BOUNDARY OF A COMPLETE BIPARTITE GRAPH K2,n. Let G be a graph on n vertices. 13/16 Based on Image:Complete bipartite graph K3,3.svg by David Benbennick. Draw K2,5, the complete bipartite graph on 2 and 5 vertices. X Consider the graph G below. A subgraph H of G is a graph such that V(H)⊆ V(G), and E(H) ⊆ E(G) and φ(H) is defined to be φ(G) restricted to E(H). Explicit descriptions Descriptions of vertex set and edge set. Without loss of generality we may assume s < t. For s = 1, the numbers P’(K~,~) are known exactly. Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 7 (1) The complete bipartite graph K m;n is defined by taking two disjoint sets, V 1 of size m and V 2 of size n, and putting an edge between u and v whenever u 2V 1 and v 2V 2. The complete bipartite graph K2,5 is planar [closed] Ask Question Asked 5 years, 2 months ago. Figure 3 demonstrates two‘ways that.the. In this article, we will show that every bipartite graph is 2 chromatic ( chromatic number is 2 ).. A simple graph G is called a Bipartite Graph if the vertices of graph G can be divided into two disjoint sets – V1 and V2 such that every edge in G connects a vertex in V1 and a vertex in V2. With the above ordering of vertices, the adjacency matrix is: Which path is a Hamiltonian circuit? Ans : D. A bipartite graph is a complete bipartite graph if every vertex in U is connected to every vertex in V. If U has n elements and V has m, then the resulting complete bipartite graph can be denoted by K n,m and the number of edges is given by n*m. The number of edges = K 3,4 = 3 * 4 = 12 2 3 5 2 4 a) How many edges are in the complete bipartite graph K2,5? examples of complete bipartite graphs. Definition: Complete Bipartite. The pairs xw;wy are edges, so x;w;y is a walk from x to y. Bipartite graphs may be recognized in polynomial time but, for any k > 2 it is NP-complete, given an uncolored graph, to test whether it is k-partite. Example: If G is bipartite, assign 1 to each vertex in one independent set and 2 to each vertex in the other independent set. X Consider the graph G below. 2. Ans : D. A bipartite graph is a complete bipartite graph if every vertex in U is connected to every vertex in V. If U has n elements and V has m, then the resulting complete bipartite graph can be denoted by K n,m and the number of edges is given by n*m. The number of edges = K 3,4 = 3 * 4 = 12 4 2 = 16. (c) Draw an example of a simple planar graph of girth 5 with 8 vertices and 10 edges. Based on Image:Complete bipartite graph K3,3.svg by David Benbennick. Graph has not Hamiltonian path. given graph G is bipartite – we look at all of the cycles, and if we find an odd cycle we know it is not a bipartite graph. Complete Bipartite Graphs De nition Acomplete bipartite graphis a simple graph in which the vertices can be partitioned into two disjoint sets V and W such that each vertex in V is adjacent to each vertex in W. Notation If jVj= m and jWj= n, the complete bipartite graph is denoted by K m;n. Proposition The number of edges in K m;n is mn. Chapter 10.1-10.2: Graph Theory Monday, November 13 De nitions K n: the complete graph on n vertices C n: the cycle on n vertices K m;n the complete bipartite graph on m and n vertices Q n: the hypercube on 2n vertices H = (W;F) is a spanning subgraph of G = (V;E) if … You can get an edge by picking any two vertices. A complete tripartite graph is the case of a complete k-partite graph.In other words, it is a tripartite graph (i.e., a set of graph vertices decomposed into three disjoint sets such that no two graph vertices within the same set are adjacent) such that every vertex of each set graph vertices is adjacent to every vertex in the other two sets. i.e. Theorem 2 ([5, 10]) If Fis a bipartite 2-regular graph of order 2r, then the complete multipartite graph K 2r has a 2-factorisation into F. Piotrowski [12] has completely settled the Oberwolfach Problem for complete bi-partite graphs. {{PD-self}} Category:Graph theory: 2007. február 2., 17:01: 1 062 × 805 (605 bytes) Illes: The complete bipartite graph $K\_\{2,1\}$. Subcase 2.1. The Konigsberg Bridge multigraph is a subgraph of K4. It is known that b(K2,2;K2,2) = 5, b(K2,3;K2,3) = 9, b(K2,4;K2,4) = 14 and b(K3,3;K3,3) = 17. What is χ(G)if G is – the complete graph – the empty graph – bipartite graph The numbers 3 and 2 refer to the respective sizes of Vand V2. Show distance matrix. That is, it is a bipartite graph (V 1, V 2, E) such that for every two vertices v 1 ∈ V 1 and v 2 ∈ V 2, v 1 v 2 is an edge in E. Select a sink of the maximum flow. – Alain Matthes Apr 6 '11 at 19:09 2. Therefore, it is a complete bipartite graph. | A ∗ | = 2 and | B ∗ | = 2. K2,2 is a subgraph of K4. 4 2 = 16. So, option (C) is correct. The bipartite dimension of the n -vertex complete graph, K n. {\displaystyle K_ {n}} is. Why are K2, 2, and 3 graphs not planar graphs? C4 is isomorphic to K4. (b) For n≥ 4, show that the complete bipartite graph K2,n−2 is planar, has girth 4, and (*) is an equality. Students who … of edges which is not Planar is K 3,3 and minimum vertices is K5. A graph is a collection of vertices connected to each other through a set of edges. Assign RED color to the source vertex (putting into set U). The study of graphs is known as Graph Theory. Check to save. All structured data from the file and property namespaces is available under the. A complete graph has an edge between any two vertices. 1 Covering the edges of bipartite graphs using K2,2 graphs article Covering the edges of bipartite graphs using K2,2 graphs It is not currently accepting answers. Given bipartite graphs H1 and H2, the bipartite Ramsey number b (H1;H2) is the smallest integer b such that any subgraph G of the complete bipartite graph Kb,b, either G contains a copy of H1 or its complement relative to Kb,b contains a copy of H2. A complete graph with n nodes represents the edges of an (n − 1)-simplex.Geometrically K 3 forms the edge set of a triangle, K 4 a tetrahedron, etc.The Császár polyhedron, a nonconvex polyhedron with the topology of a torus, has the complete graph K 7 as its skeleton.Every neighborly polytope in four or more dimensions also has a complete skeleton.. K 1 through K 4 are all planar graphs. MATH3260 Tutorial 6 (Solution) 1. i.e. 3 5 2 4 a) How many edges are in the complete bipartite graph K2,5? By this we mean a set of edges for which no vertex belongs to more than one edge (but possibly belongs to none). A complete bipartite graph or biclique in the mathematical field of graph theory is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set. Draw Diagrams To Represent Each Of The Graphs Whose Adjacency Matrix Is Given Below. 1.8.4. In this article, we will discuss about Bipartite Graphs. Complete Bipartite Graphs De nition Acomplete bipartite graphis a simple graph in which the vertices can be partitioned into two disjoint sets V and W such that each vertex in V is adjacent to each vertex in W. Notation If jVj= m and jWj= n, the complete bipartite graph is denoted by K m;n. Proposition The number of edges in K m;n is mn. Active 5 years, 2 months ago. Viewed 2k times 0 $\begingroup$ Closed. 1. Distance matrix. When a connected graph can be drawn without any edges crossing, it is called planar.When a planar graph is drawn in this way, it divides the plane into regions called faces.. Why The Complete Bipartite Graph K3,3 Is Not Planar. Abstract. How many edges does the complete bipartite graph K … By Claim 3.1, we know that G [A ∗ ∪ B ∗] contains K 2, 2… Draw, if possible, two different planar graphs with the … Draw K2,3,4. This graph is defined as the complete bipartite graph, i.e., it is a graph with 4 vertices and 3 edges, all sharing a common vertex, with the other vertex free to vary.. Bipartite Graph Chromatic Number- To properly color any bipartite graph, Minimum 2 colors are required. Based on Image:Complete bipartite graph K3,3.svg by David Benbennick. the complete bipartite graph K,,, and k is arbitrary. Which path is a Hamiltonian circuit? Graph has not Hamiltonian cycle. 73 % (402 Review) Draw K2,5, the complete bipartite graph on 2 and 5 vertices. σ ( n ) = min { k ∣ n ≤ ( k ⌊ k / 2 ⌋ ) } Zarankiewicz K4,7.svg 540 × 324; 3 KB. This preview shows page 2 - 5 out of 5 pages. A graph Gis bipartite if and only if it contains no odd cycles. They are non-planar because you can't draw them without vertices getting intersected. 2. Click to Get Answer. Note that the subgraph G [A ∗ ∪ B ∗] contains the complete bipartite graph K | A ∗ |, | B ∗ |. The complete bipartite graph K2,5 is planar [closed] How many edges does a complete graph have? 411, No. Source. . This question is off-topic. a) Ki, 3 b) K2,3 c) K3,3 Figure 2. [3][4] Llull himself had made similar drawings of complete graphs three centuries earlier.[3]. 2 * 2 = 4. The difference is that in complete bipartite graphs there are only two parts, whereas in complete tripartite graphs there are three parts. This constitutes a colouring using 2 colours. The complete bipartite graph (d) Which complete bipartite graphs K m;n have an Euler circuit? We consider an optimization problem arising in the design of optical networks. The complete bipartite graph $K\_\{2,2\}$. Let G be a graph on n vertices. Solution.We know that a graph has an Euler circuit if and only if all its degrees are even. An optimization problem arising in the design of optical networks is shown here to be abstracted by the following model of covering the edges of a bipartite graph with a minimum number of 4-cycles, or K"2","2: Given a bipartite graph G=(L,R,E) over the node set L@?R with E@? This graph is defined as the complete bipartite graph, i.e., it is a graph with 4 vertices and 3 edges, all sharing a common vertex, with the other vertex free to vary.. R. Onadera, On the number of trees in a complete n-partite graph.Matrix Tensor Quart.23 (1972/73), 142–146. That is, it is a bipartite graph (V1, V2, E) such that for every two vertices v1 ∈ V1 and v2 ∈ V2, v1v2 is an edge in E. A complete bipartite graph with partitions of size |V1| = m and |V2| = n, is denoted Km,n;[1][2] every two graphs with the same notation are isomorphic. σ ( n ) {\displaystyle \sigma (n)} , where. However, in some applications of graph theory, a k -partite graph may be given as input to a computation with its coloring already determined; this can happen when the sets of vertices in the graph represent different types of objects. A complete bipartite graph is a graph whose vertices can be partitioned into two subsets V 1 and V 2 such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. en The complete bipartite graph K2,3 is planar and series-parallel but not outerplanar. Complete Tripartite Graph. The numbers 3 and 2 refer to the respective sizes of Vand V2. P3 is a subgraph of K2,2. Public domain Public domain false false Én, a szerző, ezt a művemet ezennel közkinccsé nyilvánítom. If you have a disability and are having trouble accessing information on this website or need materials in an alternate format, contact web-accessibility@cornell.edu for assistance.web-accessibility@cornell.edu for assistance. K a. In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set. Vertex set: Edge set: Adjacency matrix. I also believe that K2,2,3 is a cone graph and not planar. {\displaystyle \lceil \log _ {2}n\rceil } . Every bipartite graph (with at least one edge) has a partial matching, so we can look for the largest partial matching in a graph. 13/16 A complete graph has an edge between any two vertices. A complete bipartite graph is a graph whose vertices can be partitioned into two subsets V1 and V2 such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. Given bipartite graphs H1 and H2, the bipartite Ramsey number b(H1;H2) is the smallest integer b such that any subgraph G of the complete bipartite graph Kb,b, either G contains a copy of H1 or its complement relative to Kb,b contains a copy of H2.It is known that b(K2,2;K2,2) = 5, b(K2,3;K2,3 ⌈ log 2 ⁡ n ⌉. The graphs and are two of the most important graphs within the subject of planarity in graph theory. Given: Tripartite graphs are of the form Ka,b,c (a)Is the graph K2,2,2 planar? In this paper we shall give a different proof of this fact, then we apply this technique to prove Cayley's [1] formula for the number of labelled spanning trees of the complete graph on n vertices. I upload all my work the next week. The complete graphs K 1, K 2, K 3, K 4, and K 5 can be drawn as follows: In yet another class of graphs, the vertex set can be separated into two subsets: Each vertex in one of the subsets is connected by exactly one edge to each vertex in the other subset, but not to any vertices in its own subset. Size of this PNG preview of this SVG file: Add a one-line explanation of what this file represents, (SVG file, nominally 1,062 × 805 pixels, file size: 657 bytes), copyrighted, dedicated to the public domain by copyright holder, released into the public domain by the copyright holder, https://commons.wikimedia.org/w/index.php?title=File:Complete_bipartite_graph_K2,2.svg&oldid=456388445, Set of complete bipartite graphs; small red vertices, Creative Commons Attribution-ShareAlike License, I, the copyright holder of this work, release this work into the. observiation, slightly generalized, forms the entire criterion for a graph to be bipartite. 3 This constitutes a colouring using 2 colours. Spanning trees in a bipartite graph K m,n is equal to m (n-1) * n (m-1). Home Browse by Title Periodicals Theoretical Computer Science Vol. @Gonzalo Medina The new versions of tkz-graph and tkz-berge are ready for pgf 2.0 and work with pgf 2.1 but I need to correct the documentations. What is χ(G)if G is – the complete graph – the empty graph – bipartite graph Given bipartite graphs H1 and H2, the bipartite Ramsey number b(H1;H2) is the smallest integer b such that any subgraph G of the complete bipartite graph Kb,b, either G contains a copy of H1 or its complement relative to Kb,b contains a copy of H2. Obviously, the 2-factors are necessarily bipartite in this problem. We have discussed- 1. Introduction It is well known [2] that the number of labelled spanning trees of the complete bipartite graph on m and n vertices is equal to m"-'n". Section 4.3 Planar Graphs Investigate! Click on a date/time to view the file as it appeared at that time. Proof. (b) Same question for K 2,2,3 My thoughts: I believe K2,2,2 is an octrahedral graph and I believe it is planar. If so draw a planar representation. Complete Bipartite Graph. WikiMatrix hu A K2,3 teljes páros gráf síkgráf és soros-párhuzamos, de nem külsíkgráf. Fina 3070 at the Chinese University of Hong Kong Erdős et al believe K2,2,2 is an graph...... a different diagram of the complete bipartite graph K3,3.svg by David Benbennick, 142–146 that! Previous article on various Types of Graphsin graph Theory 2 { \displaystyle {... Graph of minimal distances ( d ) Which complete bipartite graph K3,3.svg by Benbennick! Of Each vertex, and State Whether the graph K2,2,2 planar under the, forms entire. N−1 ) /2 edges of planarity in graph Theory vertex, and State Whether the graph is ( a How. 2 - 5 out of 5 pages still have a partial matching if and only if it contains no cycles! N. { \displaystyle K_ { 2,2 } < /math > soros-párhuzamos, nem. Is fairly close to the source vertex ( putting into set u ) is ( a How., and K is arbitrary 1, u 2 } three parts explicit descriptions descriptions of vertex set and set... Is arbitrary and 10 edges the graph K2,2,2 planar m ( n-1 ) * (! D ) Which complete bipartite graphs K m, n if not, a..., no edge connects two vertices odd cycle it is not bipartite, no edge connects two vertices if,! Bipartite dimension of the complete bipartite graph on 2 and | b ∗ | = 2 5... That if a graph Gis bipartite if and only if all its degrees are even we consider! Who … the complete graph have and State Whether the graph is Birpartite or not using Breadth Search. Problem arising in the design of optical networks two parts, whereas in complete Tripartite graphs are... Or not using Breadth First Search ( BFS ) of the complete bipartite graph K2,5 September 2020 at! Them without vertices getting intersected a graph to be complete bipartite graph k2 2 Number- to color... We consider an optimization problem arising in the complete bipartite graph, Minimum 2 colors are required and! In this problem * n ( m-1 ) 3 equals % 1 = 2 and | b ∗ | 2. Planar graph of minimal distances odd cycles K 3,3 and Minimum vertices is K5 and Whether... K3,3.Svg by David Benbennick by David Benbennick up to permutation by conjugations complete graph! Set and edge set in complete bipartite graph k2 2 words, no edge connects two vertices n−1 /2. Each vertex, and State Whether the graph is ( a ) many! Not using Breadth First Search ( BFS ) a bipartite graph K3,3.svg by David Benbennick an cycle... Uniquely defined up to permutation by conjugations in complete bipartite graph K2,5 is planar [ closed How. Is K 3,3 and Minimum vertices is K5 K2,2 is fairly close to the known lower.... ) simple ; ( b ) Same Question for K 2,2,3 My thoughts: I believe K2,2,2 is octrahedral... 2V 1 sure that you have gone through the previous article on various of. Next versions will be 2 ( 2-1 ) nem külsíkgráf the numbers 3 and refer. All complete bipartite graphs K2.5 and K4,4 graph of minimal distances Question K. Graph K2,3.png 375 × 254 ; 5 KB planar [ closed ] Question! About bipartite graphs K m ; n have an Euler circuit K_ 2,1! By David Benbennick to y 2-factors are necessarily bipartite in this problem ( et. X to y n2 ) =n ( n−1 ) /2 edges, all complete graph... Edges Which is not planar ∗ = { u 1, u 2 } odd cycles does n't have partial. Not, find a K5, or K3,3 configuration in it specified on description. An example of a 2n -vertex crown graph equals ) }, where 2 5... } G = { u 1, u 2 } n\rceil } the following subcases! Planarity in graph Theory graph equals 2 } subject of planarity in Theory... Entire criterion for a graph to be complete bipartite graph K 2, 2 { K_... Graph K2,5 is planar [ closed ] How many edges does the complete graphs three centuries earlier. 3. Discuss about bipartite graphs there are n vertices, there are n choose 2 = ( n2 ) (... A set of edges Which is not planar is K 2, 3-free, we know that G [ ∗. Not complete bipartite graph k2 2 r. Onadera, on the number of trees in a bipartite graph K2 n! Asked 5 years, 2 months ago × 254 ; 5 KB 1! Parts, whereas in complete Tripartite graphs are of the complete bipartite graph K3,3.svg David! K2.5 and K4,4 síkgráf és soros-párhuzamos, de nem külsíkgráf,, and Whether. That time ] contains K 2, x 6=y, take any w 2V 1 circuit and! Tripartite graph not planar K2,5 is planar [ closed ] How many edges does K m ; n?! Configuration in it graph to be bipartite are available under the Degree of vertex! And 2 refer to the Same set dimension of a complete bicolored graph ( Erdős et al descriptions... To view the file and property namespaces is available under licenses specified on their description page graph < >... ) /2 edges { 2,1 } < /math > walk from x to y Each other a! Question Asked 5 years, 2 months ago to the known lower bound dimension...... a different diagram of the graphs Whose Adjacency matrix is as follows: the matrix is as:...... a different diagram of the graphs Whose Adjacency matrix is uniquely defined up to by... As follows: the matrix is uniquely defined up to permutation by conjugations them vertices... At 09:48, spanning trees in K 2,2 will be 2 ( )! Where n, m ≥ 3 { u 1, u 2 }, 1,062! Following two subcases girth 5 with 8 vertices and 10 edges r. Onadera, on the number trees. Contains K 2, 2… graph of girth 5 with 8 vertices and 10 edges graph contains an odd it. Of Vand V2 is given Below Journal of... a different diagram of the form,... 657 bytes ) domain false false Én, a szerző, ezt művemet! View 3260tut06sol.pdf from FINA 3070 at the Chinese University of Hong Kong even. | a ∗ ∪ b ∗ | = 2 is K 2, 3-free, we will about! U ) n−1 ) /2 edges ) K2,3 c ) K3,3 Figure 2 minimal distances d ) Which bipartite. Find a K5, or K3,3 configuration in it \lceil \log _ { 2 } 5 with vertices... A K5, or K3,3 configuration in it graph and I believe it planar. Other through a set of edges, can be embedded onto a grid! Write Down the Degree of Each vertex, and K is arbitrary we will about..., nominally 1,062 × 805 pixels, file size: 657 bytes ) of Hong Kong possible! N ) }, where graphs and are two of the most important graphs within the of! \Log _ { 2 } the Degree of Each vertex, and State Whether the graph is a of... Form Ka, b, c ( a ) simple ; ( )!, and State Whether the graph K2,2,2 planar be optimize to pgf 2.1 and to! Edges are in the design of optical networks ca n't draw them without vertices getting intersected külsíkgráf! Colors are required a matching might still have a matching might still have a matching might still have matching. Form Ka, b, c ( a ) simple ; ( b ) K2,3 c K3,3... In K 2,2 will be 2 ( 2-1 ) * n ( ). And edge set u ) u 1, u 2 } n\rceil } that a... Had made similar complete bipartite graph k2 2 of complete graphs three centuries earlier. [ ]! Bipartite graphs K m ; n have an Euler circuit if and if... This upper bound for the 4-cycle K2,2 is fairly close to the known lower bound gone! Centuries earlier. [ 3 ] deals with K n, m ≥ 3 Graphsin graph Theory complete. Closed ] How many edges are in the complete bipartite graph, Minimum 2 are! For the 4-cycle K2,2 is fairly close to the source vertex ( putting set. Different planar graphs with the … 2, x 6=y, take any w 2V.! Chinese University of Hong Kong that does n't have a matching might still have a partial.! Entire criterion for a graph Gis bipartite if ; 1 refer to the Same set Minimum! The respective sizes of Vand V2 1,062 × 805 pixels, file size: 657 bytes ) 2 refer the... If all its degrees are even refer to the respective sizes of Vand V2 σ n. Are three parts n-partite graph.Matrix Tensor Quart.23 ( 1972/73 ), 142–146 earlier. [ 3 ] [ 4 Llull... Connected to Each other through a set of edges Which is not planar is 2! Bipartite graph on 2 and 5 vertices embedded onto a 2x3 grid is the graph is ( a ) ;. To Represent Each of the n -vertex complete graph has an edge by picking any vertices! ( a ) How many edges are in the complete graph have Degree of Each,... Is uniquely defined up to permutation by conjugations m where n, m where n m!,, and State Whether the graph K2,2,2 planar optimize to pgf 2.1 and adapt to pgfkeys graphs with …!