# The Realizability of Theta Graphs as Reconfiguration Graphs of Minimum Independent Dominating Sets

## TL;DR

This paper studies which theta graphs can be represented as reconfiguration graphs of minimum independent dominating sets in other graphs.

## Contribution

The paper characterizes theta graphs that are i-graph realizable and shows there are only finitely many exceptions.

## Key findings

- Only finitely many theta graphs are not i-graph realizable.
- All 3-connected cubic bipartite planar graphs are i-graphs.
- Characterizations of line graphs and claw-free graphs that are i-graphs are provided.

## Abstract

The independent domination number i(G) of a graph G is the minimum cardinality of a maximal independent set of G, also called an i(G)-set. The i-graph of G, denoted ℐ (G), is the graph whose vertices correspond to the i(G)-sets, and where two i(G)-sets are adjacent if and only if they differ by two adjacent vertices. Not all graphs are i-graph realizable, that is, given a target graph H, there does not necessarily exist a source graph G such that H ≅ ℐ (G). We consider a class of graphs called “theta graphs”: a theta graph is the union of three internally disjoint nontrivial paths with the same two distinct end vertices. We characterize theta graphs that are i-graph realizable, showing that there are only finitely many that are not. We also characterize those line graphs and claw-free graphs that are i-graphs, and show that all 3-connected cubic bipartite planar graphs are i-graphs.