Mathematics > Combinatorics
[Submitted on 12 Aug 2022]
Title:Mengerian graphs: characterization and recognition
View PDFAbstract:A temporal graph ${\cal G}$ is a graph that changes with time. More specifically, it is a pair $(G, \lambda)$ where $G$ is a graph and $\lambda$ is a function on the edges of $G$ that describes when each edge $e\in E(G)$ is active. Given vertices $s,t\in V(G)$, a temporal $s,t$-path is a path in $G$ that traverses edges in non-decreasing time; and if $s,t$ are non-adjacent, then a temporal $s,t$-cut is a subset $S\subseteq V(G)\setminus\{s,t\}$ whose removal destroys all temporal $s,t$-paths.
It is known that Menger's Theorem does not hold on this context, i.e., that the maximum number of internally vertex disjoint temporal $s,t$-paths is not necessarily equal to the minimum size of a temporal $s,t$-cut. In a seminal paper, Kempe, Kleinberg and Kumar (STOC'2000) defined a graph $G$ to be Mengerian if equality holds on $(G,\lambda)$ for every function $\lambda$. They then proved that, if each edge is allowed to be active only once in $(G,\lambda)$, then $G$ is Mengerian if and only if $G$ has no gem as topological minor. In this paper, we generalize their result by allowing edges to be active more than once, giving a characterization also in terms of forbidden structures. We additionally provide a polynomial time recognition algorithm.
Current browse context:
math.CO
References & Citations
Bibliographic and Citation Tools
Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)
Code, Data and Media Associated with this Article
alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)
Demos
Recommenders and Search Tools
Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.