129,587 research outputs found
Graph theory and tolerance graphs.
Graphs are diagrams made up of nodes and edges. The nodes are the points on the graph. The edges are the lines connecting the nodes. These graphs are useful in that they allow for the modeling of real world problems into a format that can be readily solved by computers. Graph theory can be used in fields as diverse as chemistry, transportation, and music. However, graph theory is not being fully utilized because of the level of knowledge required to use it. The first of three goals of this thesis is to make graph theory accessible to a larger audience by developing a graphical application. This application allows a user to create a graph, apply a graph algorithm, and display the results through a graphical user interface. The second goal of this thesis is to implement the ost useful graph algorithms. This includes basic algorithms that have been well researched and can be solved in polynomial time. Advanced algorithms for the class of graphs known as perfect graphs will also be implemented. The third goal is to add to graph theory to make it more practical. A relatively new class of graphs known as tolerance graphs allows for variations that occur in real world problems. The nodes on a tolerance graph correspond to intervals on a real line. Each interval has a tolerance value. Edges are drawn between two nodes if the intersection of the two intervals is greater than the tolerance of either interval. This thesis examines known algorithms for tolerance graphs. There are still some open problems dealing with tolerance graphs. Among them are the problem of recognizing tolerance graphs and converting a known tolerance graph into a tolerance representation. These two problems will be explored within this thesis
Perfect Roman Domination and Unique Response Roman Domination
The idea of enumeration algorithms with polynomial delay is to polynomially
bound the running time between any two subsequent solutions output by the
enumeration algorithm. While it is open for more than four decades if all
minimal dominating sets of a graph can be enumerated in output-polynomial time,
it has recently been proven that pointwise-minimal Roman dominating functions
can be enumerated even with polynomial delay. The idea of the enumeration
algorithm was to use polynomial-time solvable extension problems. We use this
as a motivation to prove that also two variants of Roman dominating functions
studied in the literature, named perfect and unique response, can be enumerated
with polynomial delay. This is interesting since Extension Perfect Roman
Domination is W[1]-complete if parameterized by the weight of the given
function and even W[2]-complete if parameterized by the number vertices
assigned 0 in the pre-solution, as we prove. Otherwise, efficient solvability
of extension problems and enumerability with polynomial delay tend to go
hand-in-hand. We achieve our enumeration result by constructing a bijection to
Roman dominating functions, where the corresponding extension problem is
polynomimaltime solvable. Furthermore, we show that Unique Response Roman
Domination is solvable in polynomial time on split graphs, while Perfect Roman
Domination is NP-complete on this graph class, which proves that both
variations, albeit coming with a very similar definition, do differ in some
complexity aspects. This way, we also solve an open problem from the
literature
The Computational Complexity of the Game of Set and its Theoretical Applications
The game of SET is a popular card game in which the objective is to form Sets
using cards from a special deck. In this paper we study single- and multi-round
variations of this game from the computational complexity point of view and
establish interesting connections with other classical computational problems.
Specifically, we first show that a natural generalization of the problem of
finding a single Set, parameterized by the size of the sought Set is W-hard;
our reduction applies also to a natural parameterization of Perfect
Multi-Dimensional Matching, a result which may be of independent interest.
Second, we observe that a version of the game where one seeks to find the
largest possible number of disjoint Sets from a given set of cards is a special
case of 3-Set Packing; we establish that this restriction remains NP-complete.
Similarly, the version where one seeks to find the smallest number of disjoint
Sets that overlap all possible Sets is shown to be NP-complete, through a close
connection to the Independent Edge Dominating Set problem. Finally, we study a
2-player version of the game, for which we show a close connection to Arc
Kayles, as well as fixed-parameter tractability when parameterized by the
number of rounds played
Protecting a Graph with Mobile Guards
Mobile guards on the vertices of a graph are used to defend it against
attacks on either its vertices or its edges. Various models for this problem
have been proposed. In this survey we describe a number of these models with
particular attention to the case when the attack sequence is infinitely long
and the guards must induce some particular configuration before each attack,
such as a dominating set or a vertex cover. Results from the literature
concerning the number of guards needed to successfully defend a graph in each
of these problems are surveyed.Comment: 29 pages, two figures, surve
Kasteleyn cokernels
We consider Kasteleyn and Kasteleyn-Percus matrices, which arise in
enumerating matchings of planar graphs, up to matrix operations on their rows
and columns. If such a matrix is defined over a principal ideal domain, this is
equivalent to considering its Smith normal form or its cokernel. Many
variations of the enumeration methods result in equivalent matrices. In
particular, Gessel-Viennot matrices are equivalent to Kasteleyn-Percus
matrices.
We apply these ideas to plane partitions and related planar of tilings. We
list a number of conjectures, supported by experiments in Maple, about the
forms of matrices associated to enumerations of plane partitions and other
lozenge tilings of planar regions and their symmetry classes. We focus on the
case where the enumerations are round or -round, and we conjecture that
cokernels remain round or -round for related ``impossible enumerations'' in
which there are no tilings. Our conjectures provide a new view of the topic of
enumerating symmetry classes of plane partitions and their generalizations. In
particular we conjecture that a -specialization of a Jacobi-Trudi matrix has
a Smith normal form. If so it could be an interesting structure associated to
the corresponding irreducible representation of \SL(n,\C). Finally we find,
with proof, the normal form of the matrix that appears in the enumeration of
domino tilings of an Aztec diamond.Comment: 14 pages, 19 in-line figures. Very minor copy correction
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