Compute highly connected subgraphs for an undirected graph

Compute highly connected subgraphs for an undirected graph

highlyConnSG(g, sat=3, ldv=c(3,2,1))
an instance of the graph class with edgemode “undirected”
singleton adoption threshold, positive integer
heuristics to remove lower degree vertice, a decreasing sequence of positive integer

A graph G with n vertices is highly connected if its connectivity k(G) > n/2. The HCS algorithm partitions a given graph into a set of highly connected subgraphs, by using minimum-cut algorithm recursively. To improve performance, the approach is refined by adopting singletons, removing low degree vertices and merging clusters.

On singleton adoption: after each round of partition, some highly connected subgraphs could be singletons (i.e., a subgraph contains only one node). To reduce the number of singletons, therefore reduce number of clusters, we try to get "normal" subgraphs to "adopt" them. If a singleton, s, has n neighbours in a highly connected subgraph c, and n > sat, we add s to c. To adapt to the modified subgraphs, this adoption process is repeated until no further such adoption.

On lower degree vertices: when the graph has low degree vertices, minimum-cut algorithm will just repeatedly separate these vertices from the rest. To reduce such expensive and non-informative computation, we "remove" these low degree vertices first before applying minimum-cut algorithm. Given ldv = (d1, d2, ..., dp), (d[i] > d[i+1] > 0), we repeat the following (i from 1 to p): remove all the highly-connected-subgraph found so far; remove vertices with degrees < di; find highly-connected-subgraphs; perform singleton adoptions.

The Boost implementation does not support self-loops, therefore we signal an error and suggest that users remove self-loops using the function removeSelfLoops function. This change does affect degree, but the original article makes no specific reference to self-loops.


A list of clusters, each is given as vertices in the graph.


A Clustering Algorithm based on Graph Connectivity by E. Hartuv, R. Shamir, 1999.

See Also

edgeConnectivity, minCut, removeSelfLoops

  • highlyConnSG
con <- file(system.file("XML/hcs.gxl",package="RBGL"))
coex <- fromGXL(con)

Documentation reproduced from package RBGL, version 1.48.1, License: Artistic-2.0

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