Graph Strength

There are several definitions of the strength of a graph.

Harary and Palmer (1959) and Harary and Palmer (1973, p. 66) define the strength of a tree as the maximum number of edges between any pair of vertices. This definition corresponds to a tree's graph diameter.

Harary and Palmer (1973, p. 117) define the strength of a multigraph as the maximum number of edges joining any two adjacent vertices.

Capobianco and Molluzzo (1979-1980) define the strength of a separable graph as , where the strength vector  of a graph is defined as the vector  of increases  in the connected component count upon deletion of vertex . For example, the Capobianco-Molluzzo strength vector of the graph illustrated above is . The Capobianco-Molluzzo strength of a nonseparable graph is then defined to be .

The most standard definition of the strength  of a simple connected graph  is

where  is the number of connected components and the minimum is taken over all edge cuts  of  (Gusfield 1983, 1991). Here, the subtraction by one in the denominator gives the number of additional connected components created. Graph strength therefore gives a measure of the resistance of a graph to edge-deletion, and so is a measure of vulnerability of a network to attack (Cunningham 1985, Gusfield 1991) and can be naturally generalized to edge-weighted graphs. Computing the strength of a graph can be done in polynomial time (Cunningham 1985, Trubin 1993).

While one could take  for disconnected graphs, using the definition of edge cuts as cuts that increase the number of connected components, the definition can be applied to give well-defined strengths for disconnected graphs.

A vertex cut analog of toughness is known as graph toughness.

The Tutte-Nash-Williams theorem states that , where  is the floor function, is the maximum number of edge-disjoint spanning trees that can be contained in a graph  (Gusfield 1984, Cunningham 1985).

REFERENCES

Capobianco, M. C. and Molluzzo, J. C. "The Strength of a Graph and Its Application to Organizational Structure." Social Networks 2, 275-283, 1979-1980.

Cunningham, W. H. "Optimal Attack and Reinforcement of a Network." J. Assoc. Comput. Mach. 32, 549-561, 1985.

Gusfield, D. "Connectivity and Edge Disjoint Spanning Trees." Inf. Proc. Lett. 16, 97-99, 1983.

Gusfield, D. "Computing the Strength of a Graph." SIAM J. Comput. 20, 639-654, 1991.

Harary, F. and Prins, G. "The Number of Homeomorphically Irreducible Trees, and Other Species." Acta Math. 101, 141-162, 1959.

Harary, F. and Palmer, E. M. Graphical Enumeration. New York: Academic Press, pp. 66 and 117, 1973.

Nash-Williams, C. St. J. A. "Edge-Disjoint Spanning Trees of Finite Graphs." J. London Math. Soc. 36, 445-450, 1961.

Schrijver, A. Combinatorial Optimization: Polyhedra and Efficiency, Vol. B. Berlin: Springer-Verlag, pp. 878 and 891, 2003.

Trubin, V. A. "Strength of a Graph and Packing of Trees and Branchings." Cyber. Syst. Anal. 29, 379-384, 1993.

Tutte, W. T. "On the Problem of Decomposing a Graph Into Connected Factors." J. London Math. Soc. 36, 221-230, 1961.