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Dynamic Processes on Complex Networks
Published in Ervin Sejdić, Tiago H. Falk, Signal Processing and Machine Learning for Biomedical Big Data, 2018
(From Ref. [31]) The graphFis an induced subgraph ofGif two vertices inFare connected if and only if they are connected inGand the vertex set and edge set ofFare subsets of the vertex set and edge set ofG. V(F)⊆V(G),E(F)⊆E(G)
Graphs from Subgraphs
Published in N. P. Shrimali, Nita H. Shah, Recent Advancements in Graph Theory, 2020
Joseph Varghese Kureethara, Johan Kok
If we remove some vertices from the existing graph, the resulting subgraph is known as an induced subgraph. If we randomly remove some edges, the resultant subgraph need not be an induced subgraph. Hence, a subgraph need not be an induced subgraph but all induced subgraphs are subgraphs. Induced subgraphs are studied extensively in the area of structural studies of graphs and networks. An excellent study on induced subgraphs is done with respect to the study of perfect graphs. A perfect graph is a graph such that for every induced subgraph of it, the clique number1 equals the chromatic number2 .
Graph Edit Distance—Theory, Algorithms, and Applications
Published in Olivier Lézoray, Leo Grady, Image Processing and Analysis with Graphs, 2012
From Definition 13.2.2 it follows that, given a graph G=(V,ℰ), a subset V′⊆V of its vertices uniquely defines a subgraph, called the subgraph induced by V′. That is, an induced subgraph of S can be obtained by removing some of its nodes (V−V′) and all their adjacent edges. However, if the second condition of Definition 13.2.2 is replaced by ℰ1⊆ℰ2 then the resulting subgraph is called noninduced. A noninduced subgraph of G is obtained by removing some of its nodes (V−V′) and all their adjacent edges plus some additional edges. An example of a graph G, an induced, and a noninduced subgraph of G is given in Figure 13.1 (note that node labels are represented using different shades of grey). Of course, given a graph G, an induced subgraph of G is also a noninduced subgraph of G.
Exploiting aggregate sparsity in second-order cone relaxations for quadratic constrained quadratic programming problems
Published in Optimization Methods and Software, 2022
Heejune Sheen, Makoto Yamashita
A vertex set is called a clique if the induced subgraph is a complete graph, and a clique C is called a maximal clique if it is not a subset of any other clique. When is a chordal graph, we can enumerate the set of maximal cliques , where p is the number of maximal cliques. Without loss of generality, we can assume that the order of is a perfect elimination ordering [14,29], and that each vertex in V is covered by at least one maximal clique, that is, .
Cooperative reference frame estimation for multi-agent systems via formation control
Published in Advanced Robotics, 2023
Consider an undirected graph with a node set and an edge set . The neighbor set of node i is denoted as Note that includes node i itself. For a node subset , the subgraph is the induced subgraph of with the node set and the edge set consisting of all of the edges in that contains the pairs of the nodes in . A node subset is called a clique if the induced subgraph is complete. A clique is said to be maximal clique if is contained by no other cliques. Let us consider the graph in Figure 2. The induced graph is complete, which implies is a clique. On the other hand, is incomplete, which implies is not a clique.
On finite-time and fixed-time consensus algorithms for dynamic networks switching among disconnected digraphs
Published in International Journal of Control, 2020
David Gómez-Gutiérrez, Carlos Renato Vázquez, Sergej Čelikovský, Juan Diego Sánchez-Torres, Javier Ruiz-León
A path from i to j in a digraph is a sequence of edges ··· starting in node i and ending in node j. A digraph is said to be connected if for every pair of vertices either there is a path from i to j or a path from j to i, otherwise it is said to be disconnected. A subgraph of is a graph such that and . A subgraph such that or is called a proper subgraph of . A subgraph of is called an induced subgraph if for any two vertices , there is an edge if and only if . An induced subgraph of that is connected is called maximal if it is not a proper subgraph of another connected subgraph of . A connected induced subgraph of that is maximal is called a connected component of . The edge connectivity of is the minimum number of edges that are needed to be removed to decrease the number of connected components.