Distance problems within Helly graphs and $k$-Helly graphs

Ducoffe, Guillaume

doi:10.48550/arxiv.2011.00001

Cited by 2 publications

(3 citation statements)

References 44 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We believe that classes of graphs with connected or G 2 -connected medians are good candidates in which the median problem can be solved faster than in O(nm) time. Our belief is based on the fact that all known such algorithms (for median graphs [20], for planar bridged triangulations [38], for Helly graphs [46], and for basis graphs of matroids [9]) use the unimodality of median functions. However, designing such minimization algorithms is not a so easy problem because they cannot use the entire distance matrix of the graph.…”

Section: Discussionmentioning

confidence: 99%

“…We used these results to compute medians in median graphs in linear time [20]. Using the unimodality of the median function on Helly graphs [9], Ducoffe [46] presented an algorithm with complexity Õ(m √ n) for computing medians of Helly graphs with n vertices and m edges. Chepoi et al [38] used unimodality of the median fuction to design a linear time algorithm for computing medians in trigraphs (planar bridged triangulations).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Graphs with $G^p$-connected medians

Bénéteau¹,

Chalopin²,

Chepoi³

et al. 2022

Preprint

View full text Add to dashboard Cite

The 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 ≥ 2, we characterize the graphs in which, with respect to any non-negative weights, median sets always induce connected subgraphs in the pth 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.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Graphs with $G^p$-connected medians

Bénéteau¹,

Chalopin²,

Chepoi³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…For instance, G = (V, E) is a k-Helly graph if every family of k-wise intersecting balls of G have a nonempty common intersection (Helly graphs are exactly the 2-Helly graphs). For every fixed k, there is a randomized Õ(m √ n)-time algorithm in order to compute the radius (minimum eccentricity of a vertex) within k-Helly graphs [46]. The Helly-gap of G = (V, E) is the least α such that, for every family of pairwise intersecting balls of G, if we increase all the radii by α then this family has a nonempty common intersection [23,42].…”

Section: Introductionmentioning

confidence: 99%

Beyond Helly graphs: the diameter problem on absolute retracts

Ducoffe¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Characterizing the graph classes such that, on n-vertex m-edge graphs in the class, we can compute the diameter faster than in 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 C if it is a retract of any G ∈ 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 Õ(m √ 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 ≥ 3. Finally, we study the diameter problem within the absolute retracts of planar graphs and split graphs, respectively.

show abstract

Distance problems within Helly graphs and $k$-Helly graphs

Cited by 2 publications

References 44 publications

Graphs with $G^p$-connected medians

Graphs with $G^p$-connected medians

Beyond Helly graphs: the diameter problem on absolute retracts

Contact Info

Product

Resources

About