In graph theory, a branch of mathematics, the splittance of an undirected graph measures its distance from a split graph. A split graph is a graph whose vertices can be partitioned into an independent set (with no edges within this subset) and a clique (having all possible edges within this subset). The splittance is the smallest number of edge additions and removals that transform the given graph into a split graph.
Calculation from degree sequence The splittance of a graph can be calculated only from the degree sequence of the graph, without examining the detailed structure of the graph. Let G be any graph with n vertices, whose degrees in decreasing order are d1 ≥ d2 ≥ d3 ≥ … ≥ dn. Let m be the largest index for which di ≥ i – 1. Then the splittance of G is
σ ( G ) = ( m 2 ) − 1 2 ∑ i = 1 m d i + 1 2 ∑ i = m + 1 n d i . {\displaystyle \sigma (G)={\tbinom {m}{2}}-{\frac {1}{2}}\sum _{i=1}^{m}d_{i}+{\frac {1}{2}}\sum _{i=m+1}^{n}d_{i}.}
The given graph is a split graph already if σ(G) = 0. Otherwise, it can be made into a split graph by calculating m, adding all missing edges between pairs of the m vertices of maximum degree, and removing all edges between pairs of the remaining vertices. As a consequence, the splittance and a sequence of edge additions and removals that realize it can be computed in linear time.
Applications The splittance of a graph has been used in parameterized complexity as a parameter to describe the efficiency of algorithms. For instance, graph coloring is fixed-parameter tractable under this parameter: it is possible to optimally color the graphs of bounded splittance in linear time.
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