6 research outputs found
Large-girth roots of graphs
We study the problem of recognizing graph powers and computing roots of
graphs. We provide a polynomial time recognition algorithm for r-th powers of
graphs of girth at least 2r+3, thus improving a bound conjectured by Farzad et
al. (STACS 2009). Our algorithm also finds all r-th roots of a given graph that
have girth at least 2r+3 and no degree one vertices, which is a step towards a
recent conjecture of Levenshtein that such root should be unique. On the
negative side, we prove that recognition becomes an NP-complete problem when
the bound on girth is about twice smaller. Similar results have so far only
been attempted for r=2,3.Comment: 14 pages, 4 figure
Square-Root Finding Problem In Graphs, A Complete Dichotomy Theorem
Graph G is the square of graph H if two vertices x,y have an edge in G if and
only if x,y are of distance at most two in H. Given H it is easy to compute its
square H^2. Determining if a given graph G is the square of some graph is not
easy in general. Motwani and Sudan proved that it is NP-complete to determine
if a given graph G is the square of some graph. The graph introduced in their
reduction is a graph that contains many triangles and is relatively dense.
Farzad et al. proved the NP-completeness for finding a square root for girth 4
while they gave a polynomial time algorithm for computing a square root of
girth at least six. Adamaszek and Adamaszek proved that if a graph has a square
root of girth six then this square root is unique up to isomorphism. In this
paper we consider the characterization and recognition problem of graphs that
are square of graphs of girth at least five. We introduce a family of graphs
with exponentially many non-isomorphic square roots, and as the main result of
this paper we prove that the square root finding problem is NP-complete for
square roots of girth five. This proof is providing the complete dichotomy
theorem for square root problem in terms of the girth of the square roots
Square Root Finding In Graphs
Abstract: Root and root finding are concepts familiar to most
branches of mathematics.
In graph theory, H is a square root of G
and G is the square of H if
two vertices x,y have an edge in G if and only if
x,y are of distance at most two in H.
Graph square is a basic operation
with a number of results about its properties in the literature.
We study the characterization
and recognition problems of graph powers.
There are algorithmic and computational approaches to answer the
decision problem of whether a given graph is a certain power of any graph.
There are polynomial time algorithms to solve this problem for square of graphs with girth at least six while the NP-completeness is proven for square of graphs with girth at most four.
The girth-parameterized problem of root fining has been open in the case of square of graphs with girth five.
We settle the conjecture that recognition of square of graphs with
girth 5 is NP-complete.
This result is providing the complete dichotomy theorem for square root finding problem
Some problems in combinatorial topology of flag complexes
In this work we study simplicial complexes associated to graphs and their homotopical and combinatorial properties. The main focus is on the family of flag complexes, which can be viewed as independence complexes and clique complexes of graphs.
In the first part we study independence complexes of graphs using two cofibre sequences corresponding to vertex and edge removals. We give applications to the connectivity of independence complexes of chordal graphs and to extremal problems in topology and we answer open questions about the homotopy types of those spaces for particular families of graphs. We also study the independence complex as a space
of configurations of particles in the so-called hard-core models on various lattices.
We define, and investigate from an algorithmic perspective, a special family of combinatorially defined homology classes in independence complexes. This enables us to give algorithms as well as NP-hardness results for topological properties of some spaces. As a corollary we prove hardness of computing homology of simplicial complexes in general.
We also view flag complexes as clique complexes of graphs. That leads to the study of various properties of Vietoris-Rips complexes of graphs.
The last result is inspired by a problem in face enumeration. Using methods of extremal graph theory we classify flag triangulations of 3-manifolds with many edges. As a corollary we complete the classification of face vectors of flag simplicial homology 3-spheres
Large-girth roots of graphs
International audienceWe study the problem of recognizing graph powers and computing roots of graphs. We provide a polynomial time recognition algorithm for r-th powers of graphs of girth at least 2r+3, thus improving a bound conjectured by Farzad et al. (STACS 2009). Our algorithm also finds all r-th roots of a given graph that have girth at least 2r+3 and no degree one vertices, which is a step towards a recent conjecture of Levenshtein that such root should be unique. On the negative side, we prove that recognition becomes an NP-complete problem when the bound on girth is about twice smaller. Similar results have so far only been attempted for r=2,3