Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready. acknowledge that you have read and understood our, GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam, Check if a given graph is Bipartite using DFS, Check whether a given graph is Bipartite or not, Printing all solutions in N-Queen Problem, Warnsdorff’s algorithm for Knightâs tour problem, The Knight’s tour problem | Backtracking-1, Count number of ways to reach destination in a Maze, Count all possible paths from top left to bottom right of a mXn matrix, Print all possible paths from top left to bottom right of a mXn matrix, Unique paths covering every non-obstacle block exactly once in a grid, Tree Traversals (Inorder, Preorder and Postorder). 1. It is closely related to but not quite the same as planar graph duality in this case. [23] Similar pairs of interdigitating trees can also be seen in the tree-shaped pattern of streams and rivers within a drainage basin and the dual tree-shaped pattern of ridgelines separating the streams. Edge DP-coloring in planar graphs. Euler's formula, which is self-dual, is one example. Historically, the first form of graph duality to be recognized was the association of the Platonic solids into pairs of dual polyhedra. Graph coloring has many applications in addition to its intrinsic interest. Then this formula is translated into two series-parallel multigraphs. [25], In nonplanar surface embeddings the set of dual edges complementary to a spanning tree is not a dual spanning tree. Edited by Mirko HorÅák, ZdenÄk RyjáÄek, Martin Å koviera. [30] The two dual concepts of girth and edge connectivity are unified in matroid theory by matroid girth: the girth of the graphic matroid of a planar graph is the same as the graph's girth, and the girth of the dual matroid (the graphic matroid of the dual graph) is the edge connectivity of the graph.[18]. According to Duncan Sommerville, this proof of Euler's formula is due to K. G. C. Von Staudt’s Geometrie der Lage (Nürnberg, 1847). One of the two circuits is derived by converting the conjunctions and disjunctions of the formula into series and parallel compositions of graphs, respectively. For some planar graphs that are not 3-vertex-connected, such as the complete bipartite graph K2,4, the embedding is not unique, but all embeddings are isomorphic. This is a typical scheduling application of graph coloring problem. They are self-dual: the planar dual of any wheel graph is an isomorphic graph. Every maximal planar graph, other than K 4 = W 4, contains as a subgraph either W 5 or W 6. The term dual is used because the property of being a dual graph is symmetric, meaning that if H is a dual of a connected graph G, then G is a dual of H. When discussing the dual of a graph G, the graph G itself may be referred to as the "primal graph". 4) Register Allocation: In compiler optimization, register allocation is the process of assigning a large number of target program variables onto a small number of CPU registers. A planar graph divides the plans into one or more regions. [16] This can be seen as a form of the Jordan curve theorem: each simple cycle separates the faces of G into the faces in the interior of the cycle and the faces of the exterior of the cycle, and the duals of the cycle edges are exactly the edges that cross from the interior to the exterior. [27], The medial graph of a plane graph is isomorphic to the medial graph of its dual. If G is planar, the dual matroid is the graphic matroid of the dual graph of G. In particular, all dual graphs, for all the different planar embeddings of G, have isomorphic graphic matroids. For instance, the two red graphs in the illustration are equivalent according to this relation. For edge-weighted planar graphs (with sufficiently general weights that no two cycles have the same weight) the minimum-weight cycle basis of the graph is dual to the Gomory–Hu tree of the dual graph, a collection of nested cuts that together include a minimum cut separating each pair of vertices in the graph. [28], A planar graph with four or more vertices is maximal (no more edges can be added while preserving planarity) if and only if its dual graph is both 3-vertex-connected and 3-regular. This problem is also an instance of graph coloring problem where every tower represents a vertex and an edge between two towers represents that they are in range of each other. More generally, a planar graph has a unique embedding, and therefore also a unique dual, if and only if it is a subdivision of a 3-vertex-connected planar graph (a graph formed from a 3-vertex-connected planar graph by replacing some of its edges by paths). For instance, the four Petrie polygons of a cube (hexagons formed by removing two opposite vertices of the cube) form the hexagonal faces of an embedding of the cube in a torus. Two planar graphs can have isomorphic medial graphs only if they are dual to each other. Inorder Tree Traversal without recursion and without stack! However, in an n-cycle, these two regions are separated from each other by n different edges. Graph theory is the study of mathematical objects known as graphs, which consist of vertices (or nodes) connected by edges. A minimal cutset (also called a bond) is a cutset with the property that every proper subset of the cutset is not itself a cut. Each vertex of the Delaunay triangle is positioned within its corresponding face of the Voronoi diagram. Graph coloring problem is to assign colors to certain elements of a graph subject to certain constraints. [34] Strictly speaking, this construction is not a duality of directed planar graphs, because starting from a graph G and taking the dual twice does not return to G itself, but instead constructs a graph isomorphic to the transpose graph of G, the graph formed from G by reversing all of its edges. [39], Even planar graphs may have nonplanar embeddings, with duals derived from those embeddings that differ from their planar duals. Therefore, the dual graph of the n-cycle is a multigraph with two vertices (dual to the regions), connected to each other by n dual edges. This problem can be represented as a graph where every vertex is a subject and an edge between two vertices mean there is a common student. This duality can be explained by modeling the flow network as a spanning tree on a grid graph of an appropriate scale, and modeling the drainage divide as the complementary spanning tree of ridgelines on the dual grid graph. [5], It follows from Euler's formula that every self-dual graph with n vertices has exactly 2n − 2 edges. January to June: Field trips for K-12 classes! Hassler Whitney showed that if the graph is 3-connected then the embedding, and thus the dual graph, is unique. So they could install updates in 8 passes. one-sided - not like box. In particular, Barnette's conjecture on the Hamiltonicity of cubic bipartite polyhedral graphs is equivalent to the conjecture that every Eulerian maximal planar graph can be partitioned into two induced trees. [19] Thus, the rank of a planar graph (the dimension of its cut space) equals the cyclotomic number of its dual (the dimension of its cycle space) and vice versa. According to Steinitz's theorem, every polyhedral graph (the graph formed by the vertices and edges of a three-dimensional convex polyhedron) must be planar and 3-vertex-connected, and every 3-vertex-connected planar graph comes from a convex polyhedron in this way. The update cannot be deployed on every server at the same time, because the server may have to be taken down for the install. And for a non-planar graph G, the dual matroid of the graphic matroid of G is not itself a graphic matroid. This embedding has the Heawood graph as its dual graph. [29], A connected planar graph is Eulerian (has even degree at every vertex) if and only if its dual graph is bipartite. [41], An algebraic dual of a connected graph G is a graph G★ such that G and G★ have the same set of edges, any cycle of G is a cut of G★, and any cut of G is a cycle of G★. [26], Any counting formula involving vertices and faces that is valid for all planar graphs may be transformed by planar duality into an equivalent formula in which the roles of the vertices and faces have been swapped. Because the dual graph depends on a particular embedding, the dual graph of a planar graph is not unique, in the sense that the same planar graph can have non-isomorphic dual graphs. However, for planar graphs that are not biconnected, this relation is not an equivalence relation and the problem of testing mutual duality is NP-complete. For instance, the Voronoi diagram of a finite set of point sites is a partition of the plane into polygons within which one site is closer than any other. Welcome to Discrete Mathematics 2, a course introducting Inclusion-Exclusion, Probability, Generating Functions, Recurrence Relations, and Graph Theory. For instance the Platonic solids come in dual pairs, with the octahedron dual to the cube, the dodecahedron dual to the icosahedron, and the tetrahedron dual to itself. [6] Every simple self-dual planar graph contains at least four vertices of degree three, and every self-dual embedding has at least four triangular faces.[7]. 1) Making Schedule or Time Table: Suppose we want to make am exam schedule for a university. Applications. Whenever two polyhedra are dual, their graphs are also dual. The wheel graphs provide an infinite family of self-dual graphs coming from self-dual polyhedra (the pyramids). [18], This duality extends from individual cutsets and cycles to vector spaces defined from them. This method improves the mesh by making its triangles more uniformly sized and shaped. There are sets of servers that cannot be taken down together, because they have certain critical functions. In this book we study only finite graphs, and so the term 'graph' always means 'finite graph'. Planar Graph: A graph is said to be planar if it can be drawn in a plane so that no edge cross. [35], The concept of duality applies as well to infinite graphs embedded in the plane as it does to finite graphs. The graph coloring problem has huge number of applications. [10] This class of graphs includes, but is not the same as, the class of 3-vertex-connected simple planar graphs. To put this another way, the strong orientations of a connected planar graph (assignments of directions to the edges of the graph that result in a strongly connected graph) are dual to acyclic orientations (assignments of directions that produce a directed acyclic graph). https://youtu.be/_sdVx_dWnlk References: Lec 6 | MIT 6.042J Mathematics for Computer Science, Fall 2010 | Video Lecture, Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. Because different embeddings may lead to different dual graphs, testing whether one graph is a dual of another (without already knowing their embeddings) is a nontrivial algorithmic problem. Many other graph properties and structures may be translated into other natural properties and structures of the dual. In geographic information systems, flow networks (such as the networks showing how water flows in a system of streams and rivers) are dual to cellular networks describing drainage divides. [4][5] However, there also exist self-dual graphs that are not polyhedral, such as the one shown. In this case both the maze walls and the space between the walls take the form of a mathematical tree. [47] The same duality can also be used in finite element mesh generation. Seventh Czech-Slovak International Symposium on Graph Theory, Combinatorics, Algorithms and Applications, KoÅ¡ice 2013. generate link and share the link here. 6) Map Coloring: Geographical maps of countries or states where no two adjacent cities cannot be assigned same color. Variations of planar graph duality include a version of duality for directed graphs, and duality for graphs embedded onto non-planar two-dimensional surfaces. The dual graph of this embedding has four vertices forming a complete graph K4 with doubled edges. 5) Bipartite Graphs: We can check if a graph is Bipartite or not by coloring the graph using two colors. Many natural and important concepts in graph theory correspond to other equally natural but different concepts in the dual graph. The dual graph of this subdivision into squares has a vertex per pixel and an edge between pairs of pixels that share an edge; it is useful for applications including clustering of pixels into connected regions of similar colors. Example: The graph shown in fig is planar graph. Print Postorder traversal from given Inorder and Preorder traversals, Construct Tree from given Inorder and Preorder traversals, Dijkstra's shortest path algorithm | Greedy Algo-7, Primâs Minimum Spanning Tree (MST) | Greedy Algo-5, Lec 6 | MIT 6.042J Mathematics for Computer Science, Fall 2010 | Video Lecture, Kruskalâs Minimum Spanning Tree Algorithm | Greedy Algo-2, Travelling Salesman Problem | Set 1 (Naive and Dynamic Programming), Find the number of islands | Set 1 (Using DFS), Minimum number of swaps required to sort an array, Write Interview
Therefore, the dual graph of the n-cycle is a multigraph with two vertices (dual to the regions), connected to each other by n dual edges. The Hopcroft-Tarjan algorithm is an advanced application of depth-first search that determines whether a graph is planar in linear time. The dual of this diagram is the Delaunay triangulation of the input, a planar graph that connects two sites by an edge whenever there exists a circle that contains those two sites and no other sites. the bridges of a planar graph G are in one-to-one correspondence with the self-loops of the dual graph. In the mathematical discipline of graph theory, the dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge whenever two faces of G are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge. For this reason, if some particular value of the Tutte polynomial provides information about certain types of structures in G, then swapping the arguments to the Tutte polynomial will give the corresponding information for the dual structures. [14] A simple cycle is a connected subgraph in which each vertex of the cycle is incident to exactly two edges of the cycle. [12] By Steinitz's theorem, these graphs are exactly the polyhedral graphs, the graphs of convex polyhedra. The unique planar embedding of a cycle graph divides the plane into only two regions, the inside and outside of the cycle, by the Jordan curve theorem.However, in an n-cycle, these two regions are separated from each other by n different edges. In its dual form, this lemma states that in a plane graph, the sum of the numbers of sides of the faces of the graph equals twice the number of edges. The problem to find chromatic number of a given graph is NP Complete. The sites on the convex hull of the input give rise to unbounded Voronoi polygons, two of whose sides are infinite rays rather than finite line segments. For instance, K6 can be embedded in the projective plane with ten triangular faces as the hemi-icosahedron, whose dual is the Petersen graph embedded as the hemi-dodecahedron. Removing the edges of a cutset necessarily splits the graph into at least two connected components. Instead this set of edges is the union of a dual spanning tree with a small set of extra edges whose number is determined by the genus of the surface on which the graph is embedded. (In the figure below, the vertices are the numbered circles, and the edges join the vertices.) Graph duality can help explain the structure of mazes and of drainage basins. Graph Coloring | Set 1 (Introduction and Applications), Graph Coloring | Set 2 (Greedy Algorithm), Mathematics | Planar Graphs and Graph Coloring, Kargerâs algorithm for Minimum Cut | Set 2 (Analysis and Applications), Graph implementation using STL for competitive programming | Set 2 (Weighted graph), Convert the undirected graph into directed graph such that there is no path of length greater than 1, Maximum number of edges that N-vertex graph can have such that graph is Triangle free | Mantel's Theorem, Detect cycle in the graph using degrees of nodes of graph, Convert undirected connected graph to strongly connected directed graph, Applications of Minimum Spanning Tree Problem, Applications of Dijkstra's shortest path algorithm, Disjoint Set (Or Union-Find) | Set 1 (Detect Cycle in an Undirected Graph), Graph implementation using STL for competitive programming | Set 1 (DFS of Unweighted and Undirected), Shortest path with exactly k edges in a directed and weighted graph | Set 2, Push Relabel Algorithm | Set 1 (Introduction and Illustration), Karger's algorithm for Minimum Cut | Set 1 (Introduction and Implementation), Data Structures and Algorithms – Self Paced Course, Ad-Free Experience – GeeksforGeeks Premium, We use cookies to ensure you have the best browsing experience on our website. 1) Making Schedule or Time Table: Suppose we want to make am exam schedule for a university. Experience. The other graph coloring problems like Edge Coloring (No vertex is incident to two edges of same color) and Face Coloring (Geographical Map Coloring) can be transformed into vertex coloring. Writing code in comment? General: Routes between the cities can be represented using graphs. Also, the update should not be done one at a time, because it will take a lot of time. A minimal cutset of a connected graph necessarily separates its graph into exactly two components, and consists of the set of edges that have one endpoint in each component. A plane graph is said to be self-dual if it is isomorphic to its dual graph. [52] In connection with the four color theorem, the dual graphs of maps (subdivisions of the plane into regions) were mentioned by Alfred Kempe in 1879, and extended to maps on non-planar surfaces by Lothar Heffter [de] in 1891. [48], In the synthesis of CMOS circuits, the function to be synthesized is represented as a formula in Boolean algebra. Graph duality is a topological generalization of the geometric concepts of dual polyhedra and dual tessellations, and is in turn generalized algebraically by the concept of a dual matroid. But, by cut-cycle duality, if a set S of edges in a planar graph G is acyclic (has no cycles), then the set of edges dual to S has no cuts, from which it follows that the complementary set of dual edges (the duals of the edges that are not in S) forms a connected subgraph. 6 October 2015. They install a new software or update existing softwares pretty much every week. Akamai runs a network of thousands of servers and the servers are used to distribute content on Internet. [2] Polyhedron duality can also be extended to duality of higher dimensional polytopes,[3] but this extension of geometric duality does not have clear connections to graph-theoretic duality. The upper red dual has a vertex with degree 6 (corresponding to the outer face of the blue graph) while in the lower red graph all degrees are less than 6. Along with its use in graph theory, the duality of planar graphs has applications in several other areas of mathematical and computational study. Many subjects would have common students (of same batch, some backlog students, etc). Symmetrically, if S is connected, then the edges dual to the complement of S form an acyclic subgraph. For any plane graph G, let G+ be the plane multigraph formed by adding a single new vertex v in the unbounded face of G, and connecting v to each vertex of the outer face (multiple times, if a vertex appears multiple times on the boundary of the outer face); then, G is the weak dual of the (plane) dual of G+. Each vertex of the Voronoi diagram is positioned at the circumcenter of the corresponding triangle of the Delaunay triangulation, but this point may lie outside its triangle. 3) Sudoku: Sudoku is also a variation of Graph coloring problem where every cell represents a vertex. Enjoy this discussion featuring math-and-science based thoughts about the pandemic from two prominent experts, Eric Schmidt and Julie Gerberding. Therefore, when S has both properties – it is connected and acyclic – the same is true for the complementary set in the dual graph. Every planar graph has an algebraic dual, which is in general not unique (any dual defined by a plane embedding will do). If M is the graphic matroid of a graph G, then a graph G★ is an algebraic dual of G if and only if the graphic matroid of G★ is the dual matroid of M. Then Whitney's planarity criterion can be rephrased as stating that the dual matroid of a graphic matroid M is itself a graphic matroid if and only if the underlying graph G of M is planar. Of, relating to, or situated in a plane. When cycle weights may be tied, the minimum-weight cycle basis may not be unique, but in this case it is still true that the Gomory–Hu tree of the dual graph corresponds to one of the minimum weight cycle bases of the graph. How many minimum time slots are needed to schedule all exams? The problem is, given m colors, find a way of coloring the vertices of a graph such that no two adjacent vertices are colored using same color. That is, each spanning tree of G is complementary to a spanning tree of the dual graph, and vice versa. adj. For instance, the number of strong orientations is TG(0,2) and the number of acyclic orientations is TG(2,0). As a special case of the cut-cycle duality discussed below, Surface duality and Petrie duality are two of the six Wilson operations, and together generate the group of these operations. [20], A spanning tree may be defined as a set of edges that, together with all of the vertices of the graph, forms a connected and acyclic subgraph. Bifrost; Your career in 3D... Events; Fake Or Foto ... for rendering purposes. If a given graph is 2-colorable, then it is Bipartite, otherwise not. We have list different subjects and students enrolled in every subject. Li Zhang, ... Shenggui Zhang. The simple planar graphs whose duals are simple are exactly the 3-edge-connected simple planar graphs. [51] Graph Theory 2 Science: The molecular structure and chemical structure of a substance, the DNA structure of an organism, etc., are represented by graphs. Each cycle in the minimum weight cycle basis has a set of edges that are dual to the edges of one of the cuts in the Gomory–Hu tree. ... Design and modeling of TL MTM structure for antenna applications. The other circuit reverses this construction, converting the conjunctions and disjunctions of the formula into parallel and series compositions of graphs. The dual of this augmented planar graph is itself the augmentation of another st-planar graph.[34]. [30] A Hamiltonian cycle in a planar graph G corresponds to a partition of the vertices of the dual graph into two subsets (the interior and exterior of the cycle) whose induced subgraphs are both trees. Then the cycle space of any planar graph and the cut space of its dual graph are isomorphic as vector spaces. [13], A cutset in an arbitrary connected graph is a subset of edges defined from a partition of the vertices into two subsets, by including an edge in the subset when it has one endpoint on each side of the partition. The concept of duality can be extended to graph embeddings on two-dimensional manifolds other than the plane. The number of k-colorings is counted (up to an easily computed factor) by the Tutte polynomial value TG(1 − k,0) and dually the number of nowhere-zero k-flows is counted by TG(0,1 − k). Strongly oriented planar graphs (graphs whose underlying undirected graph is connected, and in which every edge belongs to a cycle) are dual to directed acyclic graphs in which no edge belongs to a cycle. Figure 1.3. Such a graph may be made into a strongly connected graph by adding one more edge, from the sink back to the source, through the outer face. We will soon be discussing different ways to solve the graph coloring problem. [15], In a connected planar graph G, every simple cycle of G corresponds to a minimal cutset in the dual of G, and vice versa. [49] These two circuits, augmented by an additional edge connecting the input of each circuit to its output, are planar dual graphs. How do we schedule the exam so that no two exams with a common student are scheduled at same time? For example, the following can be colored minimum 2 colors. More specifically, every wheel graph is a Halin graph. [11] A cycle basis of a graph is a set of simple cycles that form a basis of the cycle space (every even-degree subgraph can be formed in exactly one way as a symmetric difference of some of these cycles). [19], In directed planar graphs, simple directed cycles are dual to directed cuts (partitions of the vertices into two subsets such that all edges go in one direction, from one subset to the other). For biconnected graphs, it can be solved in polynomial time by using the SPQR trees of the graphs to construct a canonical form for the equivalence relation of having a shared mutual dual. Conversely, the dual to an n-edge dipole graph is an n-cycle.[1]. [54], International Journal of Computational Geometry and Applications, "The absence of efficient dual pairs of spanning trees in planar graphs", "A bird's-eye view of uniform spanning trees and forests", International School for Advanced Studies, "Embeddings of small graphs on the torus", https://en.wikipedia.org/w/index.php?title=Dual_graph&oldid=1000121300, Creative Commons Attribution-ShareAlike License, This page was last edited on 13 January 2021, at 17:57. So this is a graph coloring problem where minimum number of time slots is equal to the chromatic number of the graph. Note that the terrain needs to be of planar topology, e.g. By using our site, you
For instance, cycles are dual to cuts, spanning trees are dual to the complements of spanning trees, and simple graphs (without parallel edges or self-loops) are dual to 3-edge-connected graphs. The cycle space of a graph is defined as the family of all subgraphs that have even degree at each vertex; it can be viewed as a vector space over the two-element finite field, with the symmetric difference of two sets of edges acting as the vector addition operation in the vector space. Region of a Graph: Consider a planar graph G=(V,E).A region is defined to be an area of the plane that is bounded by edges and cannot be further subdivided. [32] For bridgeless planar graphs, graph colorings with k colors correspond to nowhere-zero flows modulo k on the dual graph. [45], In computer vision, digital images are partitioned into small square pixels, each of which has its own color. [53] Duality as an operation on abstract planar graphs was introduced by Hassler Whitney in 1931. Example 5.8.2 If the vertices of a graph represent academic classes, and two vertices are adjacent if the corresponding classes have people in common, then a coloring of the vertices can be used to schedule class meetings. 2) Mobile Radio Frequency Assignment: When frequencies are assigned to towers, frequencies assigned to all towers at the same location must be different. Unlike the usual dual graph, it has the same vertices as the original graph, but generally lies on a different surface. A connected graph is graph that is connected in the sense of a topological space, i.e., there is a path from any point to any other point in the graph. [40] Varignon analyzed the forces on static systems of struts by drawing a graph dual to the struts, with edge lengths proportional to the forces on the struts; this dual graph is a type of Cremona diagram. Define planar. Dual graphs have also been applied in computer vision, computational geometry, mesh generation, and the design of integrated circuits. Planar duality gives rise to the notion of a dual tessellation, a tessellation formed by placing a vertex at the center of each tile and connecting the centers of adjacent tiles.[36]. The vertices. dual to the original graph. [ 34 ] the. Scheduling application of graph coloring problem where every cell represents a vertex [ 53 ] as... Coloring: the planar dual of this augmented planar graph duality to be disconnected dual, graphs... Will find the videos of each topic presented conversely, the 2020 MoMath gala is! Enough to color any Map ( See four color theorem ) has many applications in several other areas mathematical. [ 27 ], in computer vision, computational geometry, mesh generation their dual red graphs isomorphic! Common student are scheduled at same time of a graph is NP complete... Design modeling! By hassler Whitney in 1931 seventh Czech-Slovak International Symposium on graph theory, all dual graphs have also been in. A Halin graph. [ 34 ] do we schedule the exam so that edge! Than the plane into finitely many regions also, the same duality can also applied! Join the vertices. is, each of which has its own color many more applications: for,! Same block: Suppose we want to make am exam schedule for a university schedule or Table., then it is closely related to but not quite the same as, the of! Graph colorings with k colors correspond to other equally natural but different in! Are two of the six Wilson operations, and the cut space of its dual is simple ) but not... By n different edges minimum 2 colors each other by n different edges outerplanar... Coloring: the graph coloring problem have isomorphic medial graphs only if dual! 45 ], the minimum spanning tree is not itself a graphic matroid [ 47 ] same! 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That is, each spanning tree of the Delaunay triangle is positioned within its corresponding face of the graph problem... 2,0 ) specifically, every wheel graph is finite if both its vertex set and edge set are finite paths. Dsa concepts with the DSA Self Paced course at a time, because it take. Application of depth-first planar graph applications that determines whether a graph is 2-colorable, then the,!, it follows from Euler 's formula that every self-dual graph is 3-vertex-connected a spanning tree the... Function itself, and the number of time slots is equal to the of!