The i-Graph and Other Variations on the γ-Graph




Teshima, Laura Elizabeth

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In graph theory, reconfiguration is concerned with relationships among solutions to a given problem. For a graph G, the γ-graph of G, G(γ), is the graph whose vertices correspond to the minimum dominating sets of G, and where two vertices of G(γ) are adjacent if and only if their corresponding dominating sets in G differ by exactly two adjacent vertices. We present several variations of the γ-graph including those using identifying codes, locating-domination, total-domination, paired-domination, and the upper domination number. For each, we show that for any graph H, there exist infinitely many graphs whose γ-graph variant is isomorphic to H. The independent domination number i(G) is the minimum cardinality of a maximal independent set of G. The i-graph of G, denoted I(G), is the graph whose vertices correspond to the i-sets of G, and where two i-sets are adjacent if and only if they differ by two adjacent vertices. In contrast to the parameters mentioned above, we show that not all graphs are i-graph realizable. We build a series of tools to show that known i-graphs can be used to construct new i-graphs and apply these results to build other classes of i-graphs, such as block graphs, hypercubes, forests, and unicyclic graphs. We determine the structure of the i-graphs of paths and cycles, and in the case of cycles, discuss the Hamiltonicity of their i-graphs. We also construct the i-graph seeds for certain classes of line graphs, a class of graphs known as theta graphs, and maximal planar graphs. In doing so, we characterize the line graphs and theta graphs that are i-graphs.



graph theory, reconfiguration, domination, independent sets