Reconfiguration graphs of shortest paths |
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Authors: | John Asplund Kossi Edoh Ruth Haas Yulia Hristova Beth Novick Brett Werner |
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Institution: | 1. Department Of Technology and Mathematics, Dalton State College, Dalton, GA 30720, USA;2. Department of Mathematics, North Carolina Agricultural and Technical State University, Greensboro, NC 27411, USA;3. Department of Mathematics, University of Hawaii at Manoa, Honolulu, HI 96822, USA;4. Smith College, Northampton, MA 01063, United States;5. Department of Mathematics and Statistics, University of Michigan - Dearborn, Dearborn, MI 48128, USA;6. Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, USA;7. Department of Mathematics, University of Colorado, Boulder Boulder, CO 80309, USA |
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Abstract: | For a graph and, the shortest path reconfiguration graph of with respect to and is denoted by . The vertex set of is the set of all shortest paths between and in . Two vertices in are adjacent, if their corresponding paths in differ by exactly one vertex. This paper examines the properties of shortest path graphs. Results include establishing classes of graphs that appear as shortest path graphs, decompositions and sums involving shortest path graphs, and the complete classification of shortest path graphs with girth 5 or greater. We include an infinite family of well structured examples, showing that the shortest path graph of a grid graph is an induced subgraph of a lattice. |
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Keywords: | Reconfiguration graphs Shortest paths Girth Grid graph |
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