10,670 research outputs found
On the Computational Complexity of Vertex Integrity and Component Order Connectivity
The Weighted Vertex Integrity (wVI) problem takes as input an -vertex
graph , a weight function , and an integer . The
task is to decide if there exists a set such that the weight
of plus the weight of a heaviest component of is at most . Among
other results, we prove that:
(1) wVI is NP-complete on co-comparability graphs, even if each vertex has
weight ;
(2) wVI can be solved in time;
(3) wVI admits a kernel with at most vertices.
Result (1) refutes a conjecture by Ray and Deogun and answers an open
question by Ray et al. It also complements a result by Kratsch et al., stating
that the unweighted version of the problem can be solved in polynomial time on
co-comparability graphs of bounded dimension, provided that an intersection
model of the input graph is given as part of the input.
An instance of the Weighted Component Order Connectivity (wCOC) problem
consists of an -vertex graph , a weight function ,
and two integers and , and the task is to decide if there exists a set
such that the weight of is at most and the weight of
a heaviest component of is at most . In some sense, the wCOC problem
can be seen as a refined version of the wVI problem. We prove, among other
results, that:
(4) wCOC can be solved in time on interval graphs,
while the unweighted version can be solved in time on this graph
class;
(5) wCOC is W[1]-hard on split graphs when parameterized by or by ;
(6) wCOC can be solved in time;
(7) wCOC admits a kernel with at most vertices.
We also show that result (6) is essentially tight by proving that wCOC cannot
be solved in time, unless the ETH fails.Comment: A preliminary version of this paper already appeared in the
conference proceedings of ISAAC 201
Three Dimensional Software Modelling
Traditionally, diagrams used in software systems modelling have been two dimensional (2D). This is probably because graphical notations, such as those used in object-oriented and structured systems modelling, draw upon the topological graph metaphor, which, at its basic form, receives little benefit from three dimensional (3D) rendering. This paper presents a series of 3D graphical notations demonstrating effective use of the third dimension in modelling. This is done by e.g., connecting several graphs together, or in using the Z co-ordinate to show special kinds of edges. Each notation combines several familiar 2D diagrams, which can be reproduced from 2D projections of the 3D model. 3D models are useful even in the absence of a powerful graphical workstation: even 2D stereoscopic projections can expose more information than a plain planar diagram
Integrity of Generalized Transformation Graphs
The values of vulnerability helps the network designers to construct such a communication network which remains stable after some of its nodes or communication links are damaged. The transformation graphs considered in this paper are taken as model of the network system and it reveals that, how network can be made more stable and strong. For this purpose the new nodes are inserted in the network. This construction of new network is done by using the definition of generalized transformation graphs of a graphs. Integrity is one of the best vulnerability parameter. In this paper, we investigate the integrity of generalized transformation graphs and their complements. Also, we find integrity of semitotal point graph of combinations of basic graphs. Finally, we characterize few graphs having equal integrity values as that of generalized transformation graphs of same structured graphs
Atom-bond-connectivity index of certain graphs
The ABC index is one of the most applicable topological graph indices and several properties of it has been studied already due to its extensive chemical applications. Several variants of it have also been defined and used for several reasons. In this paper, we calculate the atom-bond connectivity index of some derived graphs such as double graphs, subdivision graphs and complements of some standard graphs.Publisher's Versio
Parameterized Complexity of Critical Node Cuts
We consider the following natural graph cut problem called Critical Node Cut
(CNC): Given a graph on vertices, and two positive integers and
, determine whether has a set of vertices whose removal leaves
with at most connected pairs of vertices. We analyze this problem in the
framework of parameterized complexity. That is, we are interested in whether or
not this problem is solvable in time (i.e., whether
or not it is fixed-parameter tractable), for various natural parameters
. We consider four such parameters:
- The size of the required cut.
- The upper bound on the number of remaining connected pairs.
- The lower bound on the number of connected pairs to be removed.
- The treewidth of .
We determine whether or not CNC is fixed-parameter tractable for each of
these parameters. We determine this also for all possible aggregations of these
four parameters, apart from . Moreover, we also determine whether or not
CNC admits a polynomial kernel for all these parameterizations. That is,
whether or not there is an algorithm that reduces each instance of CNC in
polynomial time to an equivalent instance of size , where
is the given parameter
Symmetry Breaking for Answer Set Programming
In the context of answer set programming, this work investigates symmetry
detection and symmetry breaking to eliminate symmetric parts of the search
space and, thereby, simplify the solution process. We contribute a reduction of
symmetry detection to a graph automorphism problem which allows to extract
symmetries of a logic program from the symmetries of the constructed coloured
graph. We also propose an encoding of symmetry-breaking constraints in terms of
permutation cycles and use only generators in this process which implicitly
represent symmetries and always with exponential compression. These ideas are
formulated as preprocessing and implemented in a completely automated flow that
first detects symmetries from a given answer set program, adds
symmetry-breaking constraints, and can be applied to any existing answer set
solver. We demonstrate computational impact on benchmarks versus direct
application of the solver.
Furthermore, we explore symmetry breaking for answer set programming in two
domains: first, constraint answer set programming as a novel approach to
represent and solve constraint satisfaction problems, and second, distributed
nonmonotonic multi-context systems. In particular, we formulate a
translation-based approach to constraint answer set solving which allows for
the application of our symmetry detection and symmetry breaking methods. To
compare their performance with a-priori symmetry breaking techniques, we also
contribute a decomposition of the global value precedence constraint that
enforces domain consistency on the original constraint via the unit-propagation
of an answer set solver. We evaluate both options in an empirical analysis. In
the context of distributed nonmonotonic multi-context system, we develop an
algorithm for distributed symmetry detection and also carry over
symmetry-breaking constraints for distributed answer set programming.Comment: Diploma thesis. Vienna University of Technology, August 201
- …