10,093 research outputs found
Acyclic Subgraphs of Planar Digraphs
An acyclic set in a digraph is a set of vertices that induces an acyclic
subgraph. In 2011, Harutyunyan conjectured that every planar digraph on
vertices without directed 2-cycles possesses an acyclic set of size at least
. We prove this conjecture for digraphs where every directed cycle has
length at least 8. More generally, if is the length of the shortest
directed cycle, we show that there exists an acyclic set of size at least .Comment: 9 page
Bipartite powers of k-chordal graphs
Let k be an integer and k \geq 3. A graph G is k-chordal if G does not have
an induced cycle of length greater than k. From the definition it is clear that
3-chordal graphs are precisely the class of chordal graphs. Duchet proved that,
for every positive integer m, if G^m is chordal then so is G^{m+2}.
Brandst\"adt et al. in [Andreas Brandst\"adt, Van Bang Le, and Thomas Szymczak.
Duchet-type theorems for powers of HHD-free graphs. Discrete Mathematics,
177(1-3):9-16, 1997.] showed that if G^m is k-chordal, then so is G^{m+2}.
Powering a bipartite graph does not preserve its bipartitedness. In order to
preserve the bipartitedness of a bipartite graph while powering Chandran et al.
introduced the notion of bipartite powering. This notion was introduced to aid
their study of boxicity of chordal bipartite graphs. Given a bipartite graph G
and an odd positive integer m, we define the graph G^{[m]} to be a bipartite
graph with V(G^{[m]})=V(G) and E(G^{[m]})={(u,v) | u,v \in V(G), d_G(u,v) is
odd, and d_G(u,v) \leq m}. The graph G^{[m]} is called the m-th bipartite power
of G.
In this paper we show that, given a bipartite graph G, if G is k-chordal then
so is G^{[m]}, where k, m are positive integers such that k \geq 4 and m is
odd.Comment: 10 page
Solving MaxSAT and #SAT on structured CNF formulas
In this paper we propose a structural parameter of CNF formulas and use it to
identify instances of weighted MaxSAT and #SAT that can be solved in polynomial
time. Given a CNF formula we say that a set of clauses is precisely satisfiable
if there is some complete assignment satisfying these clauses only. Let the
ps-value of the formula be the number of precisely satisfiable sets of clauses.
Applying the notion of branch decompositions to CNF formulas and using ps-value
as cut function, we define the ps-width of a formula. For a formula given with
a decomposition of polynomial ps-width we show dynamic programming algorithms
solving weighted MaxSAT and #SAT in polynomial time. Combining with results of
'Belmonte and Vatshelle, Graph classes with structured neighborhoods and
algorithmic applications, Theor. Comput. Sci. 511: 54-65 (2013)' we get
polynomial-time algorithms solving weighted MaxSAT and #SAT for some classes of
structured CNF formulas. For example, we get algorithms for
formulas of clauses and variables and size , if has a linear
ordering of the variables and clauses such that for any variable occurring
in clause , if appears before then any variable between them also
occurs in , and if appears before then occurs also in any clause
between them. Note that the class of incidence graphs of such formulas do not
have bounded clique-width
Some Triangulated Surfaces without Balanced Splitting
Let G be the graph of a triangulated surface of genus . A
cycle of G is splitting if it cuts into two components, neither of
which is homeomorphic to a disk. A splitting cycle has type k if the
corresponding components have genera k and g-k. It was conjectured that G
contains a splitting cycle (Barnette '1982). We confirm this conjecture for an
infinite family of triangulations by complete graphs but give counter-examples
to a stronger conjecture (Mohar and Thomassen '2001) claiming that G should
contain splitting cycles of every possible type.Comment: 15 pages, 7 figure
On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings
We study two variants of the well-known orthogonal drawing model: (i) the
smooth orthogonal, and (ii) the octilinear. Both models form an extension of
the orthogonal, by supporting one additional type of edge segments (circular
arcs and diagonal segments, respectively).
For planar graphs of max-degree 4, we analyze relationships between the graph
classes that can be drawn bendless in the two models and we also prove
NP-hardness for a restricted version of the bendless drawing problem for both
models. For planar graphs of higher degree, we present an algorithm that
produces bi-monotone smooth orthogonal drawings with at most two segments per
edge, which also guarantees a linear number of edges with exactly one segment.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
String Breaking from Ladder Diagrams in SYM Theory
The AdS/CFT correspondence establishes a string representation for Wilson
loops in N=4 SYM theory at large N and large 't Hooft coupling. One of the
clearest manifestations of the stringy behaviour in Wilson loop correlators is
the string-breaking phase transition. It is shown that resummation of planar
diagrams without internal vertices predicts the strong-coupling phase transtion
in exactly the same setting in which it arises from the string representation.Comment: 15 pages, 5 figures; v2: misprint in eq. (3.9) corrected; v4:
treatment of inhomogeneous term in the Dyson equation modifie
JaxoDraw: A graphical user interface for drawing Feynman diagrams. Version 2.0 release notes
A new version of the Feynman graph plotting tool JaxoDraw is presented.
Version 2.0 is a fundamental re-write of most of the JaxoDraw core and some
functionalities, in particular importing graphs, are not backward-compatible
with the 1.x branch. The most prominent new features include: drawing of Bezier
curves for all particle modes, on-the-fly update of edited objects, multiple
undo/redo functionality, the addition of a plugin infrastructure, and a general
improved memory performance. A new LaTeX style file is presented that has been
written specifically on top of the original axodraw.sty to meet the needs of
this this new version.Comment: 17 pages, 1 figur
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