algebravic/resolving

Find minimal resolving set for the hypercube

Use MAXSAT to find minimal resolving sets of a graph

Let $G$ be a connected undirected graph, and let $d(u,v)$ denote the minimum distance between nodes $u$ and $v$. We say that a vertex $x \in V(G)$ resolves the pair $(u,v) \in V(G) \times V(G)$ if $d(u,x) \ne d(v,x)$. A resolving set for $G$ is a subset $S \subseteq V(G)$ such that every pair of distinct vertices is resolved by some element of $S$. We would like to find a minimum cardinality resolving set. We may use MAXSAT for this. Define variables $x_s$ for all $s \in V(G)$ as an indicator of being in the minimal resolving set. Define sets $A_{u,v} = \{s \in V(G): s \text{ resolves } (u,v)\}$. Then the clauses are $\bigvee_{x \in A _ {u,v} x_s}$. We may also want to cut down the search by using automorphisms of $G$. Because automorphisms leave the distance invariant, if $S$ is a resolving set, and $\sigma \in \text{Aut}(G)$ then $\sigma(S)$ is also resolving. Let $R \subset \text{Aut}(G)$. Fix a total order on subsets of $V(G)$ (such as lexicographic order on the indicator vector). We would like any solution to satisfy $S \le \sigma(S)$ for all $\sigma \in R$.