On k-Total Dominating Graphs

TL;DR

This paper studies the properties and realizability of k-total dominating graphs, focusing on their connectivity, bounds, and specific cases like cycles and paths, to understand reconfiguration of total dominating sets.

Contribution

It characterizes the realizability of graphs as total dominating graphs and determines bounds and specific values of d_{0}(G) for various graph classes.

Findings

Any graph without isolated vertices is an induced subgraph of a graph with connected D_{k}^{t}(G).

d_{0}(G) is between the upper total domination number and the order of G.

Explicit values of d_{0}(C_{n}) and d_{0}(P_{n}) are determined.

Abstract

For a graph G, the k-total dominating graph D_{k}^{t}(G) is the graph whose vertices correspond to the total dominating sets of G that have cardinality at most k; two vertices of D_{k}^{t}(G) are adjacent if and only if the corresponding total dominating sets of G differ by either adding or deleting a single vertex. The graph D_{k}^{t}(G) is used to study the reconfiguration problem for total dominating sets: a total dominating set can be reconfigured to another by a sequence of single vertex additions and deletions, such that the intermediate sets of vertices at each step are total dominating sets, if and only if they are in the same component of D_{k}^{t}(G). Let d_{0}(G) be the smallest integer r such that D_{k}^{t}(G) is connected for all k greater than or equal to r. We investigate the realizability of graphs as total dominating graphs. For k the upper total domination number {\Gamma}_{t}(G), we show that any graph without isolated vertices is an induced subgraph of a graph G such that D_{k}^{t}(G) is connected. We show that d_{0}(G) lies between {\Gamma}_{t}(G) and n (inclusive) for any connected graph G of order n at least 3, characterize the graphs for which either bound is realized, and determine d_{0}(C_{n}) and d_{0}(P_{n}).