Location-domination and matching in cubic graphs

A dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex outside $D$ is adjacent to a vertex in $D$. A locating-dominating set of $G$ is a dominating set $D$ of $G$ with the additional property that every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u) \cap D \neq N(v) \cap D$ where $N(u)$ denotes the open neighborhood of $u$. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-domination number of $G$, denoted $γ_L(G)$, is the minimum cardinality of a locating-dominating set in $G$. Garijo, Gonzalez and Marquez [Applied Math. Computation 249 (2014), 487--501] posed the conjecture that for $n$ sufficiently large, the maximum value of the location-domination number of a twin-free, connected graph on $n$ vertices is equal to $\lfloor \frac{n}{2} \rfloor$. We propose the related (stronger) conjecture that if $G$ is a twin-free graph of order $n$ without isolated vertices, then $γ_L(G)\leq \frac{n}{2}$. We prove the conjecture for cubic graphs. We rely heavily on proof techniques from matching theory to prove our result.

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