# Asteroidal triples¶

This module contains the following function:

 is_asteroidal_triple_free() Test if the input graph is asteroidal triple-free

## Definition¶

Three independent vertices of a graph form an asteroidal triple if every two of them are connected by a path avoiding the neighborhood of the third one. A graph is asteroidal triple-free (AT-free, for short) if it contains no asteroidal triple [LB62].

Use graph_classes.AT_free.description() to get some known properties of AT-free graphs, or visit this page.

## Algorithm¶

This module implements the Straightforward algorithm recalled in [Koh04] and due to [LB62] for testing if a graph is AT-free or not. This algorithm has time complexity in $$O(n^3)$$ and space complexity in $$O(n^2)$$.

This algorithm uses the connected structure of the graph, stored into a $$n\times n$$ matrix $$M$$. This matrix is such that $$M[u][v]==0$$ if $$v\in (\{u\}\cup N(u))$$, and otherwise $$M[u][v]$$ is the unique identifier (a strictly positive integer) of the connected component of $$G\setminus(\{u\}\cup N(u))$$ to which $$v$$ belongs. This connected structure can be computed in time $$O(n(n+m))$$ using $$n$$ BFS.

Now, a triple $$u, v, w\in V$$ is an asteroidal triple if and only if it satisfies $$M[u][v]==M[u][w]$$ and $$M[v][u]==M[v][w]$$ and $$M[w][u]==M[w][v]$$, assuming all these values are positive. Indeed, if $$M[u][v]==M[u][w]$$, $$v$$ and $$w$$ are in the same connected component of $$G\setminus(\{u\}\cup N(u))$$, and so there is a path between $$v$$ and $$w$$ avoiding the neighborhood of $$u$$. The algorithm iterates over all triples.

## References¶

 [Koh04] E. Kohler. Recognizing graphs without asteroidal triples. Journal of Discrete Algorithms 2(4):439-452, Dec. 2004 doi:10.1016/j.jda.2004.04.005
 [LB62] (1, 2) C. G. Lekkerkerker, J. Ch. Boland. Representation of a finite graph by a set of intervals on the real line. Fundamenta Mathematicae, 51:45-64, 1962.

## Functions¶

sage.graphs.asteroidal_triples.is_asteroidal_triple_free(G, certificate=False)

Test if the input graph is asteroidal triple-free

An independent set of three vertices such that each pair is joined by a path that avoids the neighborhood of the third one is called an asteroidal triple. A graph is asteroidal triple-free (AT-free) if it contains no asteroidal triples. See the module's documentation for more details.

This method returns True is the graph is AT-free and False otherwise.

INPUT:

• G – a Graph
• certificate – boolean (default: False); by default, this method returns True if the graph is asteroidal triple-free and False otherwise. When certificate==True, this method returns in addition a list of three vertices forming an asteroidal triple if such a triple is found, and the empty list otherwise.

EXAMPLES:

The complete graph is AT-free, as well as its line graph:

sage: G = graphs.CompleteGraph(5)
sage: G.is_asteroidal_triple_free()
True
sage: G.is_asteroidal_triple_free(certificate=True)
(True, [])
sage: LG = G.line_graph()
sage: LG.is_asteroidal_triple_free()
True
sage: LLG = LG.line_graph()
sage: LLG.is_asteroidal_triple_free()
False


The PetersenGraph is not AT-free:

sage: from sage.graphs.asteroidal_triples import *
sage: G = graphs.PetersenGraph()
sage: G.is_asteroidal_triple_free()
False
sage: G.is_asteroidal_triple_free(certificate=True)
(False, [0, 2, 6])