# Arboricity and Subgraph Listing Algorithms

@article{Chiba1985ArboricityAS, title={Arboricity and Subgraph Listing Algorithms}, author={Norishige Chiba and Takao Nishizeki}, journal={SIAM J. Comput.}, year={1985}, volume={14}, pages={210-223} }

In this paper we introduce a new simple strategy into edge-searching of a graph, which is useful to the various subgraph listing problems. Applying the strategy, we obtain the following four algorithms. The first one lists all the triangles in a graph G in $O(a(G)m)$ time, where m is the number of edges of G and $a(G)$ the arboricity of G. The second finds all the quadrangles in $O(a(G)m)$ time. Since $a(G)$ is at most three for a planar graph G, both run in linear time for a planar graph. The…

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## References

SHOWING 1-8 OF 8 REFERENCES

### An approximation algorithm for the hamiltonian walk problem on maximal planar graphs

- MathematicsDiscret. Appl. Math.
- 1983

### Finding a minimum circuit in a graph

- MathematicsSTOC '77
- 1977

Finding minimum circuits in graphs and digraphs is discussed and an algorithm to find an almost minimum circuit is presented and an alternative method is to reduce the problem of finding a minimum circuit to that of finding an auxiliary graph.

### An algorithm for finding a large independent set in planar graphs

- MathematicsNetworks
- 1983

An O(n2) algorithm for finding an independent set that contains more than two-ninth of the vertices of a planar graph is given.

### An Analysis of Some Graph Theoretical Cluster Techniques

- Computer ScienceJACM
- 1970

Several graph theoretic cluster techniques aimed at the automatic generation of thesauri for information retrieval systems are explored and two algorithms have been tested that find maximal complete subgraphs.

### On approximating a vertex cover for planar graphs

- Mathematics, Computer ScienceSTOC '82
- 1982

The approximation problem for vertex cover of n-vertex planar graphs is treated and two results are presented: a linear time approximation algorithm and an O(n log n) time approximation scheme.

### PAPADIMITRIOU AND M. YANNAKAKIS, The clique problem for planar graphs, Inform

- Proc. Lett. 13,
- 1981

### Bounds on backtrack algorithmmsfor listing cycles

- paths, and spanning trees, Networks, 5 (1975), pp. 237-252. I11] S. TSUKIYAMA, M. IDE, I. ARIYOSHI AND I. SHIRAKAWA, A new algorithm for generating all the maximal independent sets, this Journal, 6,
- 1977