1,212 research outputs found
On the Complexity of the Interlace Polynomial
We consider the two-variable interlace polynomial introduced by Arratia,
Bollobas and Sorkin (2004). We develop graph transformations which allow us to
derive point-to-point reductions for the interlace polynomial. Exploiting these
reductions we obtain new results concerning the computational complexity of
evaluating the interlace polynomial at a fixed point. Regarding exact
evaluation, we prove that the interlace polynomial is #P-hard to evaluate at
every point of the plane, except on one line, where it is trivially polynomial
time computable, and four lines, where the complexity is still open. This
solves a problem posed by Arratia, Bollobas and Sorkin (2004). In particular,
three specializations of the two-variable interlace polynomial, the
vertex-nullity interlace polynomial, the vertex-rank interlace polynomial and
the independent set polynomial, are almost everywhere #P-hard to evaluate, too.
For the independent set polynomial, our reductions allow us to prove that it is
even hard to approximate at any point except at 0.Comment: 18 pages, 1 figure; new graph transformation (adding cycles) solves
some unknown points, error in the statement of the inapproximability result
fixed; a previous version has appeared in the proceedings of STACS 200
Interlace Polynomial of a Special Eulerian Graph
In a recent paper, Arratia, Bollobas and Sorkin discussed a graph polynomial defined recursively, which they call the interlace polynomial. There have been previous results on the interlace polynomials for special graphs, such as paths, cycles, and trees. Applications have been found in biology and other areas. In this research, I focus on the interlace polynomial of a special type of Eulerian graph, built from one cycle of size n and n cycle three graphs. I developed explicit formulas by implementing the toggling process to the graph. I further investigate the coefficients and special values of the interlace polynomial. Some of them can describe properties of the considered graph. Aigner and Holst also defined a new interlace polynomial, called the Q-interlace polynomial, recursively, which can tell other properties of the original graph. One immediate application of the Q-interlace polynomial is that a special value of it is the number of general induced subgraphs with an odd number of general perfect matchings. Thus by evaluating the Q-interlace polynomial at a specific value, we determine the number of general induced subrgaphs with an odd number of general perfect matchings of the considered Eulerian graph
Weighted interlace polynomials
The interlace polynomials introduced by Arratia, Bollobas and Sorkin extend
to invariants of graphs with vertex weights, and these weighted interlace
polynomials have several novel properties. One novel property is a version of
the fundamental three-term formula
q(G)=q(G-a)+q(G^{ab}-b)+((x-1)^{2}-1)q(G^{ab}-a-b) that lacks the last term. It
follows that interlace polynomial computations can be represented by binary
trees rather than mixed binary-ternary trees. Binary computation trees provide
a description of that is analogous to the activities description of the
Tutte polynomial. If is a tree or forest then these "algorithmic
activities" are associated with a certain kind of independent set in . Three
other novel properties are weighted pendant-twin reductions, which involve
removing certain kinds of vertices from a graph and adjusting the weights of
the remaining vertices in such a way that the interlace polynomials are
unchanged. These reductions allow for smaller computation trees as they
eliminate some branches. If a graph can be completely analyzed using
pendant-twin reductions then its interlace polynomial can be calculated in
polynomial time. An intuitively pleasing property is that graphs which can be
constructed through graph substitutions have vertex-weighted interlace
polynomials which can be obtained through algebraic substitutions.Comment: 11 pages (v1); 20 pages (v2); 27 pages (v3); 26 pages (v4). Further
changes may be made before publication in Combinatorics, Probability and
Computin
Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth
We consider the multivariate interlace polynomial introduced by Courcelle
(2008), which generalizes several interlace polynomials defined by Arratia,
Bollobas, and Sorkin (2004) and by Aigner and van der Holst (2004). We present
an algorithm to evaluate the multivariate interlace polynomial of a graph with
n vertices given a tree decomposition of the graph of width k. The best
previously known result (Courcelle 2008) employs a general logical framework
and leads to an algorithm with running time f(k)*n, where f(k) is doubly
exponential in k. Analyzing the GF(2)-rank of adjacency matrices in the context
of tree decompositions, we give a faster and more direct algorithm. Our
algorithm uses 2^{3k^2+O(k)}*n arithmetic operations and can be efficiently
implemented in parallel.Comment: v4: Minor error in Lemma 5.5 fixed, Section 6.6 added, minor
improvements. 44 pages, 14 figure
On the Complexity of the Interlace Polynomial
We consider the two-variable interlace polynomial introduced by
Arratia, Bollob`as and Sorkin (2004). We develop two graph
transformations which allow us to derive point-to-point reductions
for the interlace polynomial. Exploiting these reductions we
obtain new results concerning the computational complexity of
evaluating the interlace polynomial at a fixed point. Regarding
exact evaluation, we prove that the interlace polynomial is #P-hard
to evaluate at every point of the plane, except at one line, where
it is trivially polynomial time computable, and four lines and two
points, where the complexity mostly is still open. This solves a
problem posed by Arratia, Bollob`as and Sorkin (2004). In
particular, we observe that three specializations of the
two-variable interlace polynomial, the vertex-nullity interlace
polynomial, the vertex-rank interlace polynomial and the
independent set polynomial, are almost everywhere #P-hard to
evaluate, too. For the independent set polynomial, our reductions
allow us to prove that it is even hard to approximate at every
point except at and~
The Interlace Polynomial
In this paper, we survey results regarding the interlace polynomial of a
graph, connections to such graph polynomials as the Martin and Tutte
polynomials, and generalizations to the realms of isotropic systems and
delta-matroids.Comment: 18 pages, 5 figures, to appear as a chapter in: Graph Polynomials,
edited by M. Dehmer et al., CRC Press/Taylor & Francis Group, LL
Interlace Polynomials of Certain Graphs
In this research, we investigated the interlace polynomials of a shell graph as well as other related graphs. A shell graph, Tn is constructed by adding edges to a cycle graph such that all vertices are adjacent to one vertex. The main results of this thesis include iterative and explicit formulas for the interlace polynomial of a shell graph, denoted q(Tn; x). A linear algebra application using the adjacency matrices of the chosen graphs is also explored
- …