Decomposition andl1-Embedding of Weakly Median Graphs

Bandelt, Hans‐Jürgen; Chepoi, Victor

doi:10.1006/eujc.1999.0377

Cited by 39 publications

(50 citation statements)

References 26 publications

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“…Since every edge of G is crossed by exactly two alternating cuts from AC(G) and these cuts are convex by previous result, an observation of [2] implies that any pair of vertices x, y of G can be separated by exactly 2d(x, y) cuts of AC (G).…”

Section: This Shows That Z (E) and Z (E) Are Strips Hence E (E) And mentioning

confidence: 57%

Distance and routing labeling schemes for non-positively curved plane graphs

Chepoi¹,

Dragan²,

Vaxès³

2006

Journal of Algorithms

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Distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the distance between any two vertices u and v can be determined efficiently (e.g., in constant or logarithmic time) by merely inspecting the labels of u and v, without using any other information. Similarly, routing labeling schemes are schemes that label the vertices of a graph with short labels in such a way that given the label of a source vertex and the label of a destination, it is possible to compute efficiently (e.g., in constant or logarithmic time) the port number of the edge from the source that heads in the direction of the destination. In this paper we show that the three major classes of nonpositively curved plane graphs enjoy such distance and routing labeling schemes using O(log 2 n) bit labels on n-vertex graphs. In constructing these labeling schemes interesting metric properties of those graphs are employed.

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Section: This Shows That Z (E) and Z (E) Are Strips Hence E (E) And mentioning

confidence: 57%

Distance and routing labeling schemes for non-positively curved plane graphs

Chepoi¹,

Dragan²,

Vaxès³

2006

Journal of Algorithms

Self Cite

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“…Unlike pseudo-median graphs, weakly median graphs are closed for the Cartesian product operation, and as both classes they are also closed for gated amalgamation. Moreover, they are a GACI class in our sense, as it is shown in the following characterization from [2]. Theorem 3.3.…”

Section: Regular Graphs Of Median-like Classesmentioning

confidence: 68%

“…On the other hand, if P is an induced subgraph of a hyperoctahedron and is not regular then note that it has at least two vertices with B. Brešar not maximum degree (these are any two vertices of this graph which are not adjacent). Finally, it is straighforward to check that the bridged graphs introduced in Theorem 3.3 also enjoy (R) (by [2,Lemma 7] there are at least two vertices of degree 2 or 3, and except for K 3 and K 4 − e all graphs of this type have at least one vertex of degree at least 4). Combining these observations with Theorem 3.2 we derive Corollary 3.4.…”

Section: Regular Graphs Of Median-like Classesmentioning

confidence: 99%

Arboreal structure and regular graphs of median-like classes

Brešar

2003

Discuss. Math. Graph Theory

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We consider classes of graphs that enjoy the following properties: they are closed for gated subgraphs, gated amalgamation and Cartesian products, and for any gated subgraph the inverse of the gate function maps vertices to gated subsets. We prove that any graph of such a class contains a peripheral subgraph which is a Cartesian product of two graphs: a gated subgraph of the graph and a prime graph minus a vertex. Therefore, these graphs admit a peripheral elimination procedure which is a generalization of analogous procedure in median graphs. We characterize regular graphs of these classes whenever they enjoy an additional property. As a corollary we derive that regular weakly median graphs are precisely Cartesian products in which each factor is a complete graph or a hyperoctahedron.

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“…Fact 1 [2]. Every plane graph in which all interior faces are triangles and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than 5 is a K 4 -free bridged graph.…”

Section: Auxiliary Factsmentioning

confidence: 99%