Clique-sum

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A clique-sum of two planar graphs and the Wagner graph, forming a K5-minor-free graph. Clique-sum.svg
A clique-sum of two planar graphs and the Wagner graph, forming a K5-minor-free graph.

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.

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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.

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. [1]

There is also a close connection between clique-sums and graph connectivity: if a graph is not (k + 1)-vertex-connected (so that there exists a set of k vertices the removal of which disconnects the graph) 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.

Application in graph structure theory

A strangulated graph, formed as a clique-sum of a maximal planar graph (yellow) and two chordal graphs (red and blue) Strangulated graph.svg
A strangulated graph, formed as a clique-sum of a maximal planar graph (yellow) and two chordal graphs (red and blue)

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 [2] was a theorem of Wagner (1937), 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. [3] The graphs in which every induced cycle of length four or greater forms a minimal separator of the graph (its removal partitions the graph into two or more disconnected components, and no subset of the cycle has the same property) are exactly the clique-sums of cliques and maximal planar graphs, again without edge deletions. [4] Johnson & McKee (1996) 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 family closed under graph 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 (vertices that may be connected to an arbitrary subset of the other vertices) and vortices (graphs with low pathwidth that replace faces of the surface embedding). [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. [6]

Generalizations

The theory of clique-sums may also be generalized from graphs to matroids. [1] Notably, Seymour's decomposition theorem characterizes the regular matroids (the matroids representable by totally unimodular matrices) as the 3-sums of graphic matroids (the matroids representing spanning trees in a graph), cographic matroids, and a certain 10-element matroid. [1] [7]

Notes

  1. 1 2 3 Lovász (2006).
  2. ↑ As credited by Kříž & Thomas (1990), who list several additional clique-sum-based characterizations of graph families.
  3. ↑ Seymour & Weaver (1984).
  4. ↑ Diestel (1987).
  5. ↑ Robertson & Seymour (2003)
  6. ↑ Demaine et al. (2004); Demaine et al. (2005); Demaine, Hajiaghayi & Kawarabayashi (2005).
  7. ↑ Seymour (1980).

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