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 third lists all the complete subgraphs $K_l $ of order l in $O(la(G)^{l - 2} m)$ time. The fourth lists all the cliques in $O(a(G)m)$ time per clique. All the algorithms require linear space. We also establish an upper bound on $a(G)$ for a graph $G:a(G) \leqq \lceil (2m + n)^{1/2} \rceil $, where n is the number of vertices in G.
[1]
Jack Minker,et al.
An Analysis of Some Graph Theoretical Cluster Techniques
,
1970,
JACM.
[2]
G. G. Stokes.
"J."
,
1890,
The New Yale Book of Quotations.
[3]
Alon Itai,et al.
Finding a minimum circuit in a graph
,
1977,
STOC '77.
[4]
Norishige Chiba,et al.
An algorithm for finding a large independent set in planar graphs
,
1983,
Networks.
[5]
Takahiro Watanabe,et al.
An approximation algorithm for the hamiltonian walk problem on maximal planar graphs
,
1983,
Discret. Appl. Math..
[6]
Frank Harary,et al.
Graph Theory
,
2016
.
[7]
Reuven Bar-Yehuda,et al.
On approximating a vertex cover for planar graphs
,
1982,
STOC '82.