In graph theory, a branch of mathematics, a 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 possibly deleting some of the clique edges. A k-clique-sum is a clique-sum in which both cliques have at most k vertices. One may also form clique-sums and k-clique-sums of more than two graphs, by repeated application of the two-graph 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.
Related concepts
Clique-sums have a close connection with treewidth: If two graphs have treewidth at most k, so does their k-clique-sum. Every tree is the 1-clique-sum of its edges. Every series-parallel graph, or more generally every graph with treewidth at most two, may be formed as a 2-clique-sum of triangles. The same type of result extends to larger values of k: every graph with treewidth at most k may be formed as a clique-sum of graphs with at most k + 1 vertices; this is necessarily a k-clique-sum. There is also a close connection between clique-sums and graph connectivity: if a graph is not -vertex-connected then it may be represented as a k-clique-sum of smaller graphs. For instance, the SPQR tree of a biconnected graph is a representation of the graph as a 2-clique-sum of its triconnected components.
Clique-sums are important in graph structure theory, where they are used to characterize certain families of graphs as the graphs formed by clique-sums of simpler graphs. The first result of this type was a theorem of, who proved that the graphs that do not have a five-vertex complete graph as a minor are the 3-clique-sums of planar graphs with the eight-vertex Wagner graph; this structure theorem can be used to show that the four color theorem is equivalent to the case k = 5 of the Hadwiger conjecture. The chordal graphs are exactly the graphs that can be formed by clique-sums of cliques without deleting any edges, and the strangulated graphs are the graphs that can be formed by clique-sums of cliques and maximal planar graphs without deleting edges. The graphs in which every induced cycle of length four or greater forms a minimal separator of the graph are exactly the clique-sums of cliques and maximal planar graphs, again without edge deletions. use the clique-sums of chordal graphs and series-parallel graphs to characterize the partial matrices having positive definite completions. It is possible to derive a clique-sum decomposition for any graph familyclosed undergraph minor operations: the graphs in every minor-closed family may be formed from clique-sums of graphs that are "nearly embedded" on surfaces of bounded genus, meaning that the embedding is allowed to omit a small number of apexes and vortices. These characterizations have been used as an important tool in the construction of approximation algorithms and subexponential-time exact algorithms for NP-completeoptimization problems on minor-closed graph families.
