134 research outputs found
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
Layout of Graphs with Bounded Tree-Width
A \emph{queue layout} of a graph consists of a total order of the vertices,
and a partition of the edges into \emph{queues}, such that no two edges in the
same queue are nested. The minimum number of queues in a queue layout of a
graph is its \emph{queue-number}. A \emph{three-dimensional (straight-line
grid) drawing} of a graph represents the vertices by points in
and the edges by non-crossing line-segments. This paper contributes three main
results:
(1) It is proved that the minimum volume of a certain type of
three-dimensional drawing of a graph is closely related to the queue-number
of . In particular, if is an -vertex member of a proper minor-closed
family of graphs (such as a planar graph), then has a drawing if and only if has O(1) queue-number.
(2) It is proved that queue-number is bounded by tree-width, thus resolving
an open problem due to Ganley and Heath (2001), and disproving a conjecture of
Pemmaraju (1992). This result provides renewed hope for the positive resolution
of a number of open problems in the theory of queue layouts.
(3) It is proved that graphs of bounded tree-width have three-dimensional
drawings with O(n) volume. This is the most general family of graphs known to
admit three-dimensional drawings with O(n) volume.
The proofs depend upon our results regarding \emph{track layouts} and
\emph{tree-partitions} of graphs, which may be of independent interest.Comment: This is a revised version of a journal paper submitted in October
2002. This paper incorporates the following conference papers: (1) Dujmovic',
Morin & Wood. Path-width and three-dimensional straight-line grid drawings of
graphs (GD'02), LNCS 2528:42-53, Springer, 2002. (2) Wood. Queue layouts,
tree-width, and three-dimensional graph drawing (FSTTCS'02), LNCS
2556:348--359, Springer, 2002. (3) Dujmovic' & Wood. Tree-partitions of
-trees with applications in graph layout (WG '03), LNCS 2880:205-217, 200
On coloring parameters of triangle-free planar -graphs
An -graph is a graph with types of arcs and types of edges. A
homomorphism of an -graph to another -graph is a vertex
mapping that preserves the adjacencies along with their types and directions.
The order of a smallest (with respect to the number of vertices) such is
the -chromatic number of .Moreover, an -relative clique of
an -graph is a vertex subset of for which no two distinct
vertices of get identified under any homomorphism of . The
-relative clique number of , denoted by , is the
maximum such that is an -relative clique of . In practice,
-relative cliques are often used for establishing lower bounds of
-chromatic number of graph families.
Generalizing an open problem posed by Sopena [Discrete Mathematics 2016] in
his latest survey on oriented coloring, Chakroborty, Das, Nandi, Roy and Sen
[Discrete Applied Mathematics 2022] conjectured that for any triangle-free planar -graph and that this
bound is tight for all .In this article, we positively settle
this conjecture by improving the previous upper bound of to , and by
finding examples of triangle-free planar graphs that achieve this bound. As a
consequence of the tightness proof, we also establish a new lower bound of for the -chromatic number for the family of triangle-free
planar graphs.Comment: 22 Pages, 5 figure
Classification of edge-critical underlying absolute planar cliques for signed graphs
International audienceA simple signed graph (G,Σ) is a simple graph G having two different types of edges, positive edges and negative edges, where Σ denotes the set of negative edges of G. A closed walk of a signed graph is positive (resp., negative) if it has even (resp., odd) number of negative edges, taking repeated edges into account. A homomorphism (resp., colored homomorphism) of a simple signed graph to another simple signed graph is a vertex-mapping that preserves adjacencies and signs of closed walks (resp., signs of edges). A simple signed graph (G,Σ) is a signed absolute clique (resp., (0,2)-absolute clique) if any homomorphism (resp., colored homomorphism) of it is an injective function, in which case G is called an underlying signed absolute clique (resp., underlying (0,2)-absolute clique). Moreover, G is edge-critical if G - e is not an underlying signed absolute clique (resp., underlying (0,2)-absolute clique) for any edge e of G. In this article, we characterize all edge-critical outerplanar underlying (0,2)-absolute cliquesand all edge-critical planar underlying signed absolute cliques. We also discuss the motivations and implications of obtaining these exhaustive lists
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