Subgraphs Cliques

Second, partition each connected component into maximal compatible subgraphs (cliques). Non-deterministically retain, for each component, a partition that has the... 

Barrow, H. G. and R. M. Burstall, "Subgraph Isomorphism, Matching Relational Structures, and Maximal Cliques," Information Processing Letters, (4), 83-84 (January 1976). 

Therefore, searching for some edge-weighted maximal common subgraph for the two complete graphs is equivalent to searching for a clique [58] in this... 

H.G. Barrow, R.M. Burstall, Subgraph Isomorphism, Matching Relational Structures and Maximal Cliques, Information Proc-cesing Letters, 4 (1976) 83-84. 

A complete subgraph of a graph is also called a clique. A clique is maximal if it is not contained in any other clique [62]. A maximum clique is the maximal clique with... 

Barrow HG, Burstall RM. Subgraphs isomorphism, matching relational structures and maximal cliques. Inf Process Lett 1976 4 83-84. 

The abstract simplicial complex Cl (G) has various names it is called a flag complex in algebraic topology, while it is called a clique complex in combinatorics, prompted by the fact that clique is another term used in graph theory for complete subgraphs. 

Figure 7 An example of a clique. Assume that comparing the atoms and interatomic distances of two molecules yields the above graph in which vertices vl-v5 are the resulting correspondence. The maximally connected subgraph vl v2 v3 denotes the presence of a 3-atom common substructure.

Fig. 1.14 Other CSNs related to that depicted in Fig. 1.7c a simple, complete CSN, b threshold CSN S >0.85) the CSN linking compounds 1 is a complete subgraph/network called a clique, and c threshold CSN (5, >0.90) while compounds 1-4 are still linked they no longer form a elique...

As noted above, the subset of compounds Cpd-l,Cpd-2,Cpd-3,Cpd-4 forms a complete subgraph of the threshold graph called a clique, i.e., Tio gs o.8s- Thus, the four compounds are all hnked in the threshold CSN, while Cpd-5 is an isolated vertex as reflected by the block diagonal stmcture of the adjacency matrix in Eq. (1.57). Because of the block diagonal stmcture, each block can be treated independently of the others, a form of dimensionahty reduction. 

If the threshold is raised, to say S, > 0.90, the subset of compounds remains linked, but the subgraph induced by tlie higher threshold no longer forms a clique and c Cpd-5, of course, remains an isolated node. In this case, the adjacency matrix simplifies to... 

If we remove either 6 or u, we need to turn the subgraph made of the e node into a complete graph, i.e. if we have n nodes (from ei to e ) we need to insert n- n— l)/2 edges, and form a clique of n nodes. 

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