1 / 9

Problem: Induced Planar Graphs

Problem: Induced Planar Graphs. Tim Hayes Mentor: Dr. Fiorini. Graph Theory: Common Definitions. Graph (G): a collection of vertices (V(G)) and edges (E(G)). Simple graph: graph without loops or multiple edges.

darren
Download Presentation

Problem: Induced Planar Graphs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. 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.

  3. Example of an Induced Planar Graph • Spaces between lines are regions, where a vertex can be placed. • If there is a boundary between two regions, there is an edge between the corresponding vertices.

  4. Definitions Continued • Isomorphism: An isomorphism from G to H is a bijection f: V(G) → V(H) such that the edge (u,v) is an element of E(G) if and only if (f(u),f(v)) is an element of E(H). • Cycle: A list of vertices v₁,… vn, such that (v(i-1), vi) is an edge and the v₁ = vn. A cycle is an odd-cycle if it involves an odd number of edges and even if it involves an even number.

  5. Graphs to be discussed… Bipartite graph: A graph is bipartite if its vertex set can be partitioned into (at most) two independent sets (subgraphs without edges). Planar graph: A graph is planar if it can be drawn in the plane without edge crossings.

  6. Question: Given any graph, can this graph be an induced planar graph? Can a non-planar graph be represented as a induced planar graph? It seems that by the construction of an induced planar graph that this is not the case. If so, by Wagner’s Theorem, then K3,3 and K5 is not isomorphic to a graph that can be obtained by zero or more edge contractions of a subgraph of the induced planar graph.

  7. No non-planar graphs… but what about… …graphs with odd cycles? Can we suggest that if a graph has an odd cycle, any graph isomorphic to it is not an induced planar graph? It seems that C(2n+1) does not have an isomorphic graph that is a planar induced graph. ? ?

  8. Odd cycles? Is it the case that for any graph, if a subgraph of this graph is isomorphic to C(2n+1) then this graph is not isomorphic to any induced planar graph? Well, all faces must have an even number of edges… what if we have an odd cycle Bipartite graphs

  9. Ok, so planar and bipartite… What about this one? ?

More Related