Medians in median graphs and their cube complexes in linear time

The median of a set of vertices $P$ of a graph $G$ is the set of all vertices $x$ of $G$ minimizing the sum of distances from $x$ to all vertices of $P$. In this paper, we present a linear time algorithm to compute medians in median graphs, improving over the existing quadratic time algorithm. We also present a linear time algorithm to compute medians in the $\ell_1$-cube complexes associated with median graphs. Median graphs constitute the principal class of graphs investigated in metric graph theory and have a rich geometric and combinatorial structure, due to their bijections with CAT(0) cube complexes and domains of event structures. Our algorithm is based on the majority rule characterization of medians in median graphs and on a fast computation of parallelism classes of edges ($\Theta$-classes or hyperplanes) via Lexicographic Breadth First Search (LexBFS). To prove the correctness of our algorithm, we show that any LexBFS ordering of the vertices of $G$ satisfies the following fellow traveler property of independent interest: the parents of any two adjacent vertices of $G$ are also adjacent. Using the fast computation of the $\Theta$-classes, we also compute the Wiener index (total distance) of $G$ in linear time and the distance matrix in optimal quadratic time

A note on deterministic zombies

"Zombies and Survivor" is a variant of the well-studied game of "Cops and Robber" where the zombies (cops) can only move closer to the survivor (robber). We consider the deterministic version of the game where a zombie can choose their path if multiple options are available. The zombie number, like the cop number, of a graph is the minimum number of zombies, or cops, required to capture the survivor. In this short note, we solve a question by Fitzpatrick et al., proving that the zombie number of the Cartesian product of two graphs is at most the sum of their zombie numbers. We also give a simple graph family with cop number $2$ and an arbitrarily large zombie number.Comment: 4 page

Medians in median graphs and their cube complexes in linear time

International audienceThe median of a set of vertices P of a graph G is the set of all vertices x of G minimizing the sum of distances from $x$ to all vertices of P. In this paper, we present a linear time algorithm to compute medians in median graphs, improving over the existing quadratic time algorithm. We also present a linear time algorithm to compute medians in the l_1-cube complexes associated with median graphs. Median graphs constitute the principal class of graphs investigated in metric graph theory and have a rich geometric and combinatorial structure, due to their bijections with CAT(0) cube complexes and domains of event structures. Our algorithm is based on the majority rule characterization of medians in median graphs and on a fast computation of parallelism classes of edges (Theta-classes or hyperplanes) via Lexicographic Breadth First Search (LexBFS). To prove the correctness of our algorithm, we show that any LexBFS ordering of the vertices of G satisfies the following \emph{fellow traveler property} of independent interest: the parents of any two adjacent vertices of G are also adjacent. Using the fast computation of the Theta-classes, we also compute the Wiener index (total distance) of G in linear time and the distance matrix in optimal quadratic time

ABC(T)-graphs: an axiomatic characterization of the median procedure in graphs with connected and G2^2-connected medians

The median function is a location/consensus function that maps any profile $\pi$ (a finite multiset of vertices) to the set of vertices that minimize the distance sum to vertices from $\pi$. The median function satisfies several simple axioms: Anonymity (A), Betweeness (B), and Consistency (C). McMorris, Mulder, Novick and Powers (2015) defined the ABC-problem for consensus functions on graphs as the problem of characterizing the graphs (called, ABC-graphs) for which the unique consensus function satisfying the axioms (A), (B), and (C) is the median function. In this paper, we show that modular graphs with $G^2$-connected medians (in particular, bipartite Helly graphs) are ABC-graphs. On the other hand, the addition of some simple local axioms satisfied by the median function in all graphs (axioms (T), and (T$_2$)) enables us to show that all graphs with connected median (comprising Helly graphs, median graphs, basis graphs of matroids and even $\Delta$-matroids) are ABCT-graphs and that benzenoid graphs are ABCT$_2$-graphs. McMorris et al (2015) proved that the graphs satisfying the pairing property (called the intersecting-interval property in their paper) are ABC-graphs. We prove that graphs with the pairing property constitute a proper subclass of bipartite Helly graphs and we discuss the complexity status of the recognition problem of such graphs

Graphs with GpG^p-connected medians

31 pages, 5 figuresThe median of a graph $G$ with weighted vertices is the set of all vertices $x$ minimizing the sum of weighted distances from $x$ to the vertices of $G$. For any integer $p\ge 2$, we characterize the graphs in which, with respect to any non-negative weights, median sets always induce connected subgraphs in the $p$th power $G^p$ of $G$. This extends some characterizations of graphs with connected medians (case $p=1$) provided by Bandelt and Chepoi (2002). The characteristic conditions can be tested in polynomial time for any $p$. We also show that several important classes of graphs in metric graph theory, including bridged graphs (and thus chordal graphs), graphs with convex balls, bucolic graphs, and bipartite absolute retracts, have $G^2$-connected medians. Extending the result of Bandelt and Chepoi that basis graphs of matroids are graphs with connected medians, we characterize the isometric subgraphs of Johnson graphs and of halved-cubes with connected medians

A note on deterministic zombies

International audienceZombies and Survivor is a variant of the well-studied game of Cops and Robbers where the zombies (cops) can only move closer to the survivor (robber). We consider the deterministic version of the game where a zombie can choose their path if multiple options are available. Similar to the cop number, the zombie number of a graph is the minimum number of zombies required to capture the survivor. In this short note, we solve a question by Fitzpatrick et al., proving that the zombie number of the Cartesian product of two graphs is at most the sum of their zombie numbers. We also give a simple graph family with cop number 2 and an arbitrarily large zombie number