181 research outputs found
On the Coloring of Pseudoknots
Pseudodiagrams are diagrams of knots where some information about which
strand goes over/under at certain crossings may be missing. Pseudoknots are
equivalence classes of pseudodiagrams, with equivalence defined by a class of
Reidemeister-type moves. In this paper, we introduce two natural extensions of
classical knot colorability to this broader class of knot-like objects. We use
these definitions to define the determinant of a pseudoknot (i.e. the
pseudodeterminant) that agrees with the classical determinant for classical
knots. Moreover, we extend Conway notation to pseudoknots to facilitate the
investigation of families of pseudoknots and links. The general formulae for
pseudodeterminants of pseudoknot families may then be used as a criterion for
p-colorability of pseudoknots.Comment: 22 pages, 24 figure
Recognizing Graph Theoretic Properties with Polynomial Ideals
Many hard combinatorial problems can be modeled by a system of polynomial
equations. N. Alon coined the term polynomial method to describe the use of
nonlinear polynomials when solving combinatorial problems. We continue the
exploration of the polynomial method and show how the algorithmic theory of
polynomial ideals can be used to detect k-colorability, unique Hamiltonicity,
and automorphism rigidity of graphs. Our techniques are diverse and involve
Nullstellensatz certificates, linear algebra over finite fields, Groebner
bases, toric algebra, convex programming, and real algebraic geometry.Comment: 20 pages, 3 figure
List coloring in the absence of a linear forest.
The k-Coloring problem is to decide whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. The Listk-Coloring problem requires in addition that every vertex u must receive a color from some given set L(u)⊆{1,…,k}. Let Pn denote the path on n vertices, and G+H and rH the disjoint union of two graphs G and H and r copies of H, respectively. For any two fixed integers k and r, we show that Listk-Coloring can be solved in polynomial time for graphs with no induced rP1+P5, hereby extending the result of Hoà ng, Kamiński, Lozin, Sawada and Shu for graphs with no induced P5. Our result is tight; we prove that for any graph H that is a supergraph of P1+P5 with at least 5 edges, already List 5-Coloring is NP-complete for graphs with no induced H
Finding Optimal Solutions With Neighborly Help
Can we efficiently compute optimal solutions to instances of a hard problem from optimal solutions to neighboring (i.e., locally modified) instances? For example, can we efficiently compute an optimal coloring for a graph from optimal colorings for all one-edge-deleted subgraphs? Studying such questions not only gives detailed insight into the structure of the problem itself, but also into the complexity of related problems; most notably graph theory\u27s core notion of critical graphs (e.g., graphs whose chromatic number decreases under deletion of an arbitrary edge) and the complexity-theoretic notion of minimality problems (also called criticality problems, e.g., recognizing graphs that become 3-colorable when an arbitrary edge is deleted).
We focus on two prototypical graph problems, Colorability and Vertex Cover. For example, we show that it is NP-hard to compute an optimal coloring for a graph from optimal colorings for all its one-vertex-deleted subgraphs, and that this remains true even when optimal solutions for all one-edge-deleted subgraphs are given. In contrast, computing an optimal coloring from all (or even just two) one-edge-added supergraphs is in P. We observe that Vertex Cover exhibits a remarkably different behavior, demonstrating the power of our model to delineate problems from each other more precisely on a structural level.
Moreover, we provide a number of new complexity results for minimality and criticality problems. For example, we prove that Minimal-3-UnColorability is complete for DP (differences of NP sets), which was previously known only for the more amenable case of deleting vertices rather than edges. For Vertex Cover, we show that recognizing beta-vertex-critical graphs is complete for Theta_2^p (parallel access to NP), obtaining the first completeness result for a criticality problem for this class
Complexity of Stability
Graph parameters such as the clique number, the chromatic number, and the
independence number are central in many areas, ranging from computer networks
to linguistics to computational neuroscience to social networks. In particular,
the chromatic number of a graph (i.e., the smallest number of colors needed to
color all vertices such that no two adjacent vertices are of the same color)
can be applied in solving practical tasks as diverse as pattern matching,
scheduling jobs to machines, allocating registers in compiler optimization, and
even solving Sudoku puzzles. Typically, however, the underlying graphs are
subject to (often minor) changes. To make these applications of graph
parameters robust, it is important to know which graphs are stable for them in
the sense that adding or deleting single edges or vertices does not change
them. We initiate the study of stability of graphs for such parameters in terms
of their computational complexity. We show that, for various central graph
parameters, the problem of determining whether or not a given graph is stable
is complete for \Theta_2^p, a well-known complexity class in the second level
of the polynomial hierarchy, which is also known as "parallel access to NP.
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