10,332 research outputs found
Characterizations of consistent marked graphs
AbstractA marked graph is a graph with a + or − sign on each vertex and is called consistent if each cycle has an even number of − signs. This concept is motivated by problems of communication networks and social networks. We present some new characterizations and recognition algorithms for consistent marked graphs
Characterization of Line-Consistent Signed Graphs
The line graph of a graph with signed edges carries vertex signs. A
vertex-signed graph is consistent if every circle (cycle, circuit) has positive
vertex-sign product. Acharya, Acharya, and Sinha recently characterized
line-consistent signed graphs, i.e., edge-signed graphs whose line graphs, with
the naturally induced vertex signature, are consistent. Their proof applies
Hoede's relatively difficult characterization of consistent vertex-signed
graphs. We give a simple proof that does not depend on Hoede's theorem as well
as a structural description of line-consistent signed graphs.Comment: 5 pages. V2 defines sign of a walk and corrects statement of Theorem
4 ("is balanced and" was missing); also minor copyeditin
Efficient Algorithms for Membership in Boolean Hierarchies of Regular Languages
The purpose of this paper is to provide efficient algorithms that decide
membership for classes of several Boolean hierarchies for which efficiency (or
even decidability) were previously not known. We develop new forbidden-chain
characterizations for the single levels of these hierarchies and obtain the
following results: - The classes of the Boolean hierarchy over level
of the dot-depth hierarchy are decidable in (previously only the
decidability was known). The same remains true if predicates mod for fixed
are allowed. - If modular predicates for arbitrary are allowed, then
the classes of the Boolean hierarchy over level are decidable. - For
the restricted case of a two-letter alphabet, the classes of the Boolean
hierarchy over level of the Straubing-Th\'erien hierarchy are
decidable in . This is the first decidability result for this hierarchy. -
The membership problems for all mentioned Boolean-hierarchy classes are
logspace many-one hard for . - The membership problems for quasi-aperiodic
languages and for -quasi-aperiodic languages are logspace many-one complete
for
Negation Switching Equivalence in Signed Graphs
Unless mentioned or defined otherwise, for all terminology and notion in graph theory the reader is refer to [8]. We consider only finite, simple graphs free from self-loops
Total Minimal Dominating Signed Graph
Cartwright and Harary considered graphs in which vertices represent persons and the edges represent symmetric dyadic relations amongst persons each of which designated as being positive or negative according to whether the nature of the relationship is positive (friendly, like, etc.) or negative (hostile, dislike, etc.). Such a network S is called a signed graph. Signed graphs are much studied in literature because of their extensive use in modeling a variety socio-psychological process and also because of their interesting connections with many classical mathematical systems
Graphs with Plane Outside-Obstacle Representations
An \emph{obstacle representation} of a graph consists of a set of polygonal
obstacles and a distinct point for each vertex such that two points see each
other if and only if the corresponding vertices are adjacent. Obstacle
representations are a recent generalization of classical polygon--vertex
visibility graphs, for which the characterization and recognition problems are
long-standing open questions.
In this paper, we study \emph{plane outside-obstacle representations}, where
all obstacles lie in the unbounded face of the representation and no two
visibility segments cross. We give a combinatorial characterization of the
biconnected graphs that admit such a representation. Based on this
characterization, we present a simple linear-time recognition algorithm for
these graphs. As a side result, we show that the plane vertex--polygon
visibility graphs are exactly the maximal outerplanar graphs and that every
chordal outerplanar graph has an outside-obstacle representation.Comment: 12 pages, 7 figure
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