148 research outputs found
The Complexity of Helly- EPG Graph Recognition
Golumbic, Lipshteyn, and Stern defined in 2009 the class of EPG graphs, the
intersection graph class of edge paths on a grid. An EPG graph is a graph
that admits a representation where its vertices correspond to paths in a grid
, such that two vertices of are adjacent if and only if their
corresponding paths in have a common edge. If the paths in the
representation have at most bends, we say that it is a -EPG
representation. A collection of sets satisfies the Helly property when
every sub-collection of that is pairwise intersecting has at least one
common element. In this paper, we show that given a graph and an integer
, the problem of determining whether admits a -EPG representation
whose edge-intersections of paths satisfy the Helly property, so-called
Helly--EPG representation, is in NP, for every bounded by a polynomial
function of . Moreover, we show that the problem of recognizing
Helly--EPG graphs is NP-complete, and it remains NP-complete even when
restricted to 2-apex and 3-degenerate graphs
Instruction Scheduling Across Control Flow
ABSTRACT Instruction scheduling algorithms are used in compilers to reduce run-time delays for the compiled code by the reordering or transformation of program statements, usually at the intermediate language or assembly code level. Considerable research has been carried out on scheduling code within the scope of basic blocks, i.e., straight line sections of code, and very effective basic block schedulers are now included in most modern compilers and especially for pipeline processors. In previous work Golumbic high quality basic block scheduler by first suppressing selected subsequences of instructions and then scheduling the modified sequence of instructions using the basic block scheduler. A candidate subsequence for suppression can be found by identifying a region of a program control flow graph, called an S-region, which has a unique entry and a unique exit and meets predetermined criteria. This enables scheduling of a sequence of instructions beyond basic block boundaries, with only minimal changes to an existing compiler, by identifying beneficial opportunities to cover delays that would otherwise have been beyond its scope
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
Representations of Edge Intersection Graphs of Paths in a Tree
Let be a collection of nontrivial simple paths in a tree . The edge intersection graph of , denoted by EPT(), has vertex set that corresponds to the members of , and two vertices are joined by an edge if the corresponding members of share a common edge in . An undirected graph is called an edge intersection graph of paths in a tree, if for some and . The EPT graphs are useful in network applications. Scheduling undirected calls in a tree or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph. It is known that recognition and coloring of EPT graphs are NP-complete problems. However, the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs, and therefore can be colored in polynomial time complexity. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. This also implies that the coloring of the edge intersection graph of paths in a degree 4 tree is polynomial. We raise a number of intriguing conjectures regarding related families of graphs
Recognizing Chordal-Bipartite Probe Graphs
A graph G is chordal-bipartite probe if its vertices can be partitioned into two sets P (probes) and N (non-probes) where N is a stable set and such that G can be extended to a chordal-bipartite graph by adding edges between non-probes. A bipartite graph is called chordal-bipartite if it contains no chordless cycle of length strictly greater than 5. Such probe/non-probe completion problems have been studied previously on other families of graphs, such as interval graphs and chordal graphs. In this paper, we give a characterization of chordal-bipartite probe graphs, in the case of a fixed given partition of the vertices into probes and nonprobes. Our results are obtained by solving first the more general case without assuming that N is a stable set, and then this can be applied to the more specific case. Our characterization uses an edge elimination ordering which also implies a polynomial time recognition algorithm for the class. This research was conducted in the context of a France-Israel Binational project, while the French team visited Haifa in March 2007
Comparability graph coloring for optimizing utilization of stream register files in stream processors
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