On the extension of vertex maps to graph homomorphisms

AbstractA reflexive graph is a simple undirected graph where a loop has been added at each vertex. If G and H are reflexive graphs and U⊆V(H), then a vertex map f:U→V(G) is called nonexpansive if for every two vertices x,y∈U, the distance between f(x) and f(y) in G is at most that between x and y in H. A reflexive graph G is said to have the extension property (EP) if for every reflexive graph H, every U⊆V(H) and every nonexpansive vertex map f:U→V(G), there is a graph homomorphism φf:H→G that agrees with f on U. Characterizations of EP-graphs are well known in the mathematics and computer science literature. In this article we determine when exactly, for a given “sink”-vertex s∈V(G), we can obtain such an extension φf;s that maps each vertex of H closest to the vertex s among all such existing homomorphisms φf. A reflexive graph G satisfying this is then said to have the sink extension property (SEP). We then characterize the reflexive graphs with the unique sink extension property (USEP), where each such sink extensions φf;s is unique

Absolutely minimal Lipschitz extension of tree-valued mappings

We prove that every Lipschitz function from a subset of a locally compact length space to a metric tree has a unique absolutely minimal Lipschitz extension (AMLE). We relate these extensions to a stochastic game called Politics—a generalization of a game called Tug of War that has been used in Peres et al. (J Am Math Soc 22(1):167–210, 2009) to study real-valued AMLEs.National Science Foundation (U.S.) (NSF Grant CCF-0832795)National Science Foundation (U.S.) (NSF Grant CCF-0635078)United States-Israel Binational Science Foundation (BSF grant 2006009)National Science Foundation (U.S.) (NSF Grant OISE-0730136)National Science Foundation (U.S.) (NSF Grant DMS-0645585

Beyond Helly graphs: the diameter problem on absolute retracts

Characterizing the graph classes such that, on $n$-vertex $m$-edge graphs in the class, we can compute the diameter faster than in ${\cal O}(nm)$ time is an important research problem both in theory and in practice. We here make a new step in this direction, for some metrically defined graph classes. Specifically, a subgraph $H$ of a graph $G$ is called a retract of $G$ if it is the image of some idempotent endomorphism of $G$. Two necessary conditions for $H$ being a retract of $G$ is to have $H$ is an isometric and isochromatic subgraph of $G$. We say that $H$ is an absolute retract of some graph class ${\cal C}$ if it is a retract of any $G \in {\cal C}$ of which it is an isochromatic and isometric subgraph. In this paper, we study the complexity of computing the diameter within the absolute retracts of various hereditary graph classes. First, we show how to compute the diameter within absolute retracts of bipartite graphs in randomized $\tilde{\cal O}(m\sqrt{n})$ time. For the special case of chordal bipartite graphs, it can be improved to linear time, and the algorithm even computes all the eccentricities. Then, we generalize these results to the absolute retracts of $k$-chromatic graphs, for every fixed $k \geq 3$. Finally, we study the diameter problem within the absolute retracts of planar graphs and split graphs, respectively

Netlike partial cubes II. Retracts and netlike subgraphs

AbstractFirst we show that the class of netlike partial cubes is closed under retracts. Then we prove, for a subgraph G of a netlike partial cube H, the equivalence of the assertions: G is a netlike subgraph of H; G is a hom-retract of H; G is a retract of H. Finally we show that a non-trivial netlike partial cube G, which is a retract of some bipartite graph H, is also a hom-retract of H if and only if G contains at most one convex cycle of length greater than 4