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Problem: Induced Planar Graphs Tim Hayes Mentor: Dr. Fiorini

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Graph Theory: Common Definitions Graph (G): a collection of vertices (V(G)) and edges (E(G)). Simple graph: graph without loops or multiple edges. Subgraph: graph H is a subgraph of graph G if V(H) is a subset of V(G) and E(H) is a subset of E(H). Connected graph: A graph that has a u,v-path for each pair of vertices u,v.

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Example of an Induced Planar Graph Let A be an arrangement of lines. The lines divide the plane into f(A) regions. An induced planar graph G(A) is a graph induced by line arrangement A with vertex set V(A) in which each planar region is represented by a distinct vertex. Two distinct vertices will be joined by an edge if and only if the corresponding regions are adjacent.

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Question: Given any graph, can this graph be an induced planar graph? First, given a graph G, can we install an arrangement of lines that corresponds with the graph? ? Second, given that the first is possible, does the arrangement of lines determine anything more outside of the graph?

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Ideas Intersection points of lines correspond with faces on a graph. Corollary 1: In an induced planar graph, if there is a cycle, within the subgraph which only contains that cycle, there is at least one intersection point. Corollary 2: In an induced planar graph, every edge involves two vertices on either side. …only planar graphs…

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Ideas (cont’d) Theorem 1.2: In an induced planar graph, if f(A)>1, it is not the case that there is a face which is not at all connected by an edge to another. Theorem 1.3: In an induced planar graph, if two faces are connected to each other by edges, it is necessarily the case that they are connected only by one edge. Theorem 1.1: In an induced planar graph, every cycle in the graph is an even cycle.

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Answer to first question Compilation of even-cycle graphs where each face is connected to another but by only one edge.

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Second Question: given that the first is possible, does the arrangement of lines determine anything more outside of the graph? Let a line be parallel to another line if it is not involved in a face with that line. Claim: In any such case, the transitive property of parallelism, parallelism in the way described, is broken. After grouping together lines that are parallel to each other, if the graph is an induced planar graph the total number of elements in all of these groups is the same number of lines there are in the arrangement. If not, the total number of elements is more than the number of lines.

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