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. If two graphs G and H each contain cliques of equal size, the clique-sum of G and H is formed from their disjoint union by identifying pairs of vertices in these two cliques to form a single shared clique, and then deleting all the clique edges (the original definition, based on the notion of set sum) or possibly deleting some of the clique edges (a loosening of the definition). A k-clique-sum is a clique-sum in which both cliques have exactly (or sometimes, at most) k vertices. One may also form clique-sums and k-clique-sums of more than two graphs, by repeated application of the clique-sum operation.
Different sources disagree on which edges should be removed as part of a clique-sum operation. In some contexts, such as the decomposition of chordal graphs or strangulated graphs, no edges should be removed. 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.