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{\bf Tobias M\"uller, Attila P\'or and Jean-S\'ebastien Sereni}
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{\bf Graphs with Four Boundary Vertices}
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A vertex $v$ of a graph $G$ is a \emph{boundary vertex} if there
exists a vertex $u$ such that the distance in $G$ from $u$ to $v$ is
at least the distance from $u$ to any neighbour of $v$. We give a
full description of all graphs that have exactly four boundary
vertices, which answers a question of Hasegawa and Saito. To this
end, we introduce the concept of frame of a graph. It allows us to
construct, for every positive integer $b$ and every possible
``distance-vector'' between $b$ points, a graph $G$ with exactly $b$
boundary vertices such that every graph with $b$ boundary vertices and
the same distance-vector between them is an induced subgraph of $G$.
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