Clique-sum

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The clique-sum of two planar graphs and the Wagner graph, forming a K5-free graph.
The clique-sum of two planar graphs and the Wagner graph, forming a K5-free graph.

In graph theory, 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.

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]

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.[3] 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).[4]. 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.[5]

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][6]

[edit] Notes

  1. ^ a b c 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. ^ Robertson & Seymour (2003)
  5. ^ Demaine et al. (2004); Demaine et al. (2005); Demaine, Hajiaghayi & Kawarabayashi (2005).
  6. ^ Seymour (1980).

[edit] References

  • Wagner, Klaus (1937), “Über eine Eigenschaft der ebenen Komplexe”, Mathematische Annalen 114: 570–590 .