4 research outputs found
Finding Small Complete Subgraphs Efficiently
(I) We revisit the algorithmic problem of finding all triangles in a graph
with vertices and edges. According to a result of Chiba and
Nishizeki (1985), this task can be achieved by a combinatorial algorithm
running in time, where is the
graph arboricity. We provide a new very simple combinatorial algorithm for
finding all triangles in a graph and show that is amenable to the same running
time analysis. We derive these worst-case bounds from first principles and with
very simple proofs that do not rely on classic results due to Nash-Williams
from the 1960s.
(II) We extend our arguments to the problem of finding all small complete
subgraphs of a given fixed size. We show that the dependency on and
in the running time of the algorithm of
Chiba and Nishizeki for listing all copies of , where , is
asymptotically tight.
(III) We give improved arboricity-sensitive running times for counting and/or
detection of copies of , for small . A key ingredient in
our algorithms is, once again, the algorithm of Chiba and Nishizeki. Our new
algorithms are faster than all previous algorithms in certain high-range
arboricity intervals for every .Comment: 14 pages, 1 figure. arXiv admin note: substantial text overlap with
arXiv:2105.0126
Efficient Algorithms for Subgraph Listing
Subgraph isomorphism is a fundamental problem in graph theory. In this paper we focus on listing subgraphs isomorphic to a given pattern graph. First, we look at the algorithm due to Chiba and Nishizeki for listing complete subgraphs of fixed size, and show that it cannot be extended to general subgraphs of fixed size. Then, we consider the algorithm due to Ga̧sieniec et al. for finding multiple witnesses of a Boolean matrix product, and use it to design a new output-sensitive algorithm for listing all triangles in a graph. As a corollary, we obtain an output-sensitive algorithm for listing subgraphs and induced subgraphs isomorphic to an arbitrary fixed pattern graph
Efficient Algorithms for Subgraph Listing
Subgraph isomorphism is a fundamental problem in graph theory. In this paper we focus on listing subgraphs isomorphic to a given pattern graph. First, we look at the algorithm due to Chiba and Nishizeki for listing complete subgraphs of fixed size, and show that it cannot be extended to general subgraphs of fixed size. Then, we consider the algorithm due to Ga̧sieniec et al. for finding multiple witnesses of a Boolean matrix product, and use it to design a new output-sensitive algorithm for listing all triangles in a graph. As a corollary, we obtain an output-sensitive algorithm for listing subgraphs and induced subgraphs isomorphic to an arbitrary fixed pattern graph
Efficient Algorithms for Subgraph Listing
Subgraph isomorphism is a fundamental problem in graph theory. In this paper we focus on listing subgraphs isomorphic to a given pattern graph. First, we look at the algorithm due to Chiba and Nishizeki for listing complete subgraphs of fixed size, and show that it cannot be extended to general subgraphs of fixed size. Then, we consider the algorithm due to Ga̧sieniec et al. for finding multiple witnesses of a Boolean matrix product, and use it to design a new output-sensitive algorithm for listing all triangles in a graph. As a corollary, we obtain an output-sensitive algorithm for listing subgraphs and induced subgraphs isomorphic to an arbitrary fixed pattern graph