Arbre (théorie des graphes)

Résumé

En théorie des graphes, un arbre est un graphe acyclique et connexe. Sa forme évoque en effet la ramification des branches d'un arbre. Par opposition aux arbres simples, arbres binaires, ou arbres généraux de l'analyse d'algorithme ou de la combinatoire analytique, qui sont des plongements particuliers d'arbres (graphes) dans le plan, on appelle parfois les arbres (graphes) arbres de Cayley, car ils sont comptés par la formule de Cayley. Un ensemble d'arbres est appelé une forêt. Définition Définition intuitive Un graphe représente un ensemble de points, appelés sommets ou nœuds, reliés ou non entre eux par des traits, appelés arêtes. Il s'agit donc d'un concept très simple et d'un outil de modélisation très général, utile dans de nombreux domaines.

Par exemple une carte routière peut être modélisée par un graphe : les villes sont les sommets et on reliera deux sommets par un trait si et seulement si les villes qu'ils représentent ont une route les conn

Source officielle

https://fr.wikipedia.org/wiki/Arbre_(th%C3%A9orie_des_graphes)

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Budgeted sensor placement for source localization on trees

Laura Elisa Celis, Brunella Marta Spinelli, Patrick Thiran

We address the problem of choosing a fixed number of sensor vertices in a graph in order to detect the source of a partially-observed diffusion process on the graph itself. Building on the definition of double resolvability we introduce a notion of vertex resolvability. For the case of tree graphs we give polynomial time algorithms for both finding the sensors that maximize the probability of correct detection of the source and for identifying the sensor set that minimizes the expected distance between the real source and the estimated one.

2015

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Parity (matching theory) and connectivity (network flows) are two main branches of combinatorial optimization. In an attempt to understand better their interrelation, we study a problem where both parity and connectivity requirements are imposed. The main result is a characterization of undirected graphs G = (V,E) having a k-edge-connected T-odd orientation for every subset with |E| + |T| even. (T-odd orientation: the in-degree of v is odd precisely if v is in T.) As a corollary, we obtain that every (2k)-edge- connected graph with |V| + |E| even has a (k-1)-edge- connected orientation in which the in-degree of every node is odd. Along the way, a structural characterization will be given for digraphs with a root-node s having k edge- disjoint paths from s to every node and k-1 edge-disjoint paths from every node to s.

2001

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Mask Combination of Multi-Layer Graphs for Global Structure Inference

Eda Bayram, Pascal Frossard, Elif Vural

Structure inference is an important task for network data processing and analysis in data science. In recent years, quite a few approaches have been developed to learn the graph structure underlying a set of observations captured in a data space. Although real-world data is often acquired in settings where relationships are influenced by a priori known rules, such domain knowledge is still not well exploited in structure inference problems. In this paper, we identify the structure of signals defined in a data space whose inner relationships are encoded by multi-layer graphs. We aim at properly exploiting the information originating from each layer to infer the global structure underlying the signals. We thus present a novel method for combining the multiple graphs into a global graph using mask matrices, which are estimated through an optimization problem that accommodates the multi-layer graph information and a signal representation model. The proposed mask combination method also estimates the contribution of each graph layer in the structure of signals. The experiments conducted both on synthetic and real-world data suggest that integrating the multi-layer graph representation of the data in the structure inference framework enhances the learning procedure considerably by adapting to the quality and the quantity of the input data.

2020

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