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\subsection*{J. Chalopin, V. Chepoi, U. Giocanti: Graphs with convex balls and groups acting on them}
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\index{Chalopin, J.}\index{Chepoi, V.}\index{Giocanti, U.}
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Jérémie Chalopin, LIS, Marseille, {\tt jeremie.chalopin@lis-lab.fr}\\
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Victor Chepoi, LIS, Marseille, {\tt victor.chepoi@lis-lab.fr}\\
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\underline{Ugo Giocanti}, G-Scop, Grenoble, {\tt ugo.giocanti@grenoble-inp.fr}\\ %souligner l'orateur
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Metric graph theory consists in studying graphs as metric spaces, when they are equipped with the shortest-path metric. During the last few decades, graph theoretic analogues of well-known geometrical properties have been intensively studied. In particular, it was shown that  \emph{systolic graphs} behave similarly to spaces with nonpositive curvature (also called CAT(0) spaces). Systolic graphs are graphs 
in which triangles are the only isometric cycles (a subgraph $H$ of a graph $G$ is isometric if the distances between vertices of $H$ coincide with their distances in $G$). They generalize in a natural way
chordal graphs, and were first studied separately in the eighties by Chepoi and Soltan \cite{SC} and by Farber and Jamison \cite{FJ}.

An important property of CAT(0) spaces and systolic graphs is that all balls are convex (a ball in a graph is the set of vertices at distance at most $d$ from some vertex). A natural problem is to characterize the graphs that satisfy this property. These graphs are called \emph{CB-graphs}, and were also introduced in the 80s (see \cite{SC}, \cite{FJ}).

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We give a new characterization of CB-graphs as those graphs satisfying two classical metric properties, called the Interval Neighborhood Condition and the Triangle Pentagon Condition. We also give a local to global characterization of CB-graphs, giving a topological insight to their structure. Finally, we give a local characterization of some special clique paths in CB-graphs, implying thanks to a result of Swiatkowski that groups
acting geometrically on CB-graphs are biautomatic (we will give all the necessary background on group actions during the talk).

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\begin{thebibliography}{99}

\bibitem{SC}
V.~Soltan and V.~Chepoi, \emph{Conditions for invariance of sets diameters under $d$-convexification in a graph}, English transl., Cybernetics {\bf 19(6)} (1983), 750--756.

\bibitem{FJ}
M.~Farber and R.~Jamison, \emph{On local convexity in graphs}, Discrete Mathematics {\bf 66(3)} (1987), 231--247.

\end{thebibliography}


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