Please use ide.geeksforgeeks.org, [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. Applications of Graph Coloring: The graph coloring problem has huge number of applications. Variations of planar graph duality include a version of duality for directed graphs, and duality for graphs embedded onto non-planar two-dimensional surfaces. [22], An example of this type of decomposition into interdigitating trees can be seen in some simple types of mazes, with a single entrance and no disconnected components of its walls. 1) Making Schedule or Time Table: Suppose we want to make am exam schedule for a university. One of the two circuits is derived by converting the conjunctions and disjunctions of the formula into series and parallel compositions of graphs, respectively. [43], For nonplanar surface embeddings, unlike planar duals, the dual graph is not generally an algebraic dual of the primal graph. Welcome to Discrete Mathematics 2, a course introducting Inclusion-Exclusion, Probability, Generating Functions, Recurrence Relations, and Graph Theory. 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. [12] By Steinitz's theorem, these graphs are exactly the polyhedral graphs, the graphs of convex polyhedra. Conversely, the dual to an n-edge dipole graph is an n-cycle.[1]. [45], In computer vision, digital images are partitioned into small square pixels, each of which has its own color. Thus, the edges of any planar graph and its dual can together be partitioned (in multiple different ways) into two spanning trees, one in the primal and one in the dual, that together extend to all the vertices and faces of the graph but never cross each other. The dual of a simple graph need not be simple: it may have self-loops (an edge with both endpoints at the same vertex) or multiple edges connecting the same two vertices, as was already evident in the example of dipole multigraphs being dual to cycle graphs. 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. If the free space of the maze is partitioned into simple cells (such as the squares of a grid) then this system of cells can be viewed as an embedding of a planar graph, in which the tree structure of the walls forms a spanning tree of the graph and the tree structure of the free space forms a spanning tree of the dual graph. [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. 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". Any scenario in which one wishes to examine the structure of a network of connected objects is potentially a problem for graph theory. [17] The girth of any planar graph (the size of its smallest cycle) equals the edge connectivity of its dual graph (the size of its smallest cutset). [21] However, this does not work for shortest path trees, even approximately: there exist planar graphs such that, for every pair of a spanning tree in the graph and a complementary spanning tree in the dual graph, at least one of the two trees has distances that are significantly longer than the distances in its graph. generate link and share the link here. Complete Graph. The wheel graphs provide an infinite family of self-dual graphs coming from self-dual polyhedra (the pyramids). Therefore, a planar graph is simple if and only if its dual has no 1- or 2-edge cutsets; that is, if it is 3-edge-connected. 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. Therefore, when S has both properties – it is connected and acyclic – the same is true for the complementary set in the dual graph. one-sided - not like box. If the graph is planar, then it must follow below Euler's Formula for planar graphs v - e + f = 2 v is number of vertices e is number of edges f is number of faces including bounded and unbounded 10 - 15 + f = 2 f = 7 There is always one unbounded face, so the number of bounded faces = 6 Every three-dimensional convex polyhedron has a dual polyhedron; the dual polyhedron has a vertex for every face of the original polyhedron, with two dual vertices adjacent whenever the corresponding two faces share an edge. This is a typical scheduling application of graph coloring problem. There can be many more applications: For example the below reference video lecture has a case study at 1:18. [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. 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. [51] According to Duncan Sommerville, this proof of Euler's formula is due to K. G. C. Von Staudt’s Geometrie der Lage (Nürnberg, 1847). Surface duality and Petrie duality are two of the six Wilson operations, and together generate the group of these operations. [10] This class of graphs includes, but is not the same as, the class of 3-vertex-connected simple planar graphs. Vertex coloring is the most common graph coloring problem. There is an edge between two vertices if they are in same row or same column or same block. 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. Many natural and important concepts in graph theory correspond to other equally natural but different concepts in the dual graph. The dual of this augmented planar graph is itself the augmentation of another st-planar graph.[34]. A planar graph divides the plans into one or more regions. [33], An st-planar graph is a connected planar graph together with a bipolar orientation of that graph, an orientation that makes it acyclic with a single source and a single sink, both of which are required to be on the same face as each other. (In the figure below, the vertices are the numbered circles, and the edges join the vertices.) We call a graph with just one vertex trivial and ail other graphs nontrivial. Hassler Whitney showed that if the graph is 3-connected then the embedding, and thus the dual graph, is unique. [40] [38], The same concept works equally well for non-orientable surfaces. Planar Graph: A graph is said to be planar if it can be drawn in a plane so that no edge cross. 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. 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. They are self-dual: the planar dual of any wheel graph is an isomorphic graph. [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. 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. Four colors are sufficient to color any map (See Four Color Theorem). How do we schedule the exam so that no two exams with a common student are scheduled at same time? Define planar. 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. 6) Map Coloring: Geographical maps of countries or states where no two adjacent cities cannot be assigned same color. That is, each spanning tree of G is complementary to a spanning tree of the dual graph, and vice versa. With this constraint, the dual of any surface-embedded graph has a natural embedding on the same surface, such that the dual of the dual is isomorphic to and isomorphically embedded to the original graph. In most applications of this concept, it is restricted to embeddings with the property that each face is a topological disk; this constraint generalizes the requirement for planar graphs that the graph be connected. Graph Theory gives us, both an easy way to pictorially represent many major mathematical results, and insights into the deep theories behind them. 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. This was even before Leonhard Euler's 1736 work on the Seven Bridges of Königsberg that is often taken to be the first work on graph theory. [14] A simple cycle is a connected subgraph in which each vertex of the cycle is incident to exactly two edges of the cycle. Such a graph is called a dipole graph. 5) Bipartite Graphs: We can check if a graph is Bipartite or not by coloring the graph using two colors. Every maximal planar graph, other than K 4 = W 4, contains as a subgraph either W 5 or W 6. 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). The extra edges, in combination with paths in the spanning trees, can be used to generate the fundamental group of the surface. [5], It follows from Euler's formula that every self-dual graph with n vertices has exactly 2n − 2 edges. Graph duality can help explain the structure of mazes and of drainage basins. Along with its use in graph theory, the duality of planar graphs has applications in several other areas of mathematical and computational study. [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. As a special case of the cut-cycle duality discussed below, Graph Coloring | Set 1 (Introduction and Applications), Graph Coloring | Set 2 (Greedy Algorithm), Mathematics | Planar Graphs and Graph Coloring, Karger’s 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. What is the minimum number of frequencies needed? A basic graph of 3-Cycle. 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. [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]. [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. Are disconnected is 3-connected then the cycle space of its dual graph, other than 4! Same concept works equally well for non-orientable surfaces using strings, not integer indices, to define refer. 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