Clique-sum

In graph theory, a branch of mathematics, a clique sum (or clique-sum) is a way of combining two graphs by gluing them together at a clique, analogous to the connected sum operation in topology.

Different sources disagree on which edges should be removed as part of a clique-sum operation.

In other contexts, such as the SPQR-tree decomposition of graphs into their 3-vertex-connected components, all edges should be removed.

And in yet other contexts, such as the graph structure theorem for minor-closed families of simple graphs, it is natural to allow the set of removed edges to be specified as part of the operation.

[5] These characterizations have been used as an important tool in the construction of approximation algorithms and subexponential-time exact algorithms for NP-complete optimization problems on minor-closed graph families.