74 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
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
Parameterized Domination in Circle Graphs
A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Applied Mathematics, 42(1):51-63, 1993] proved that Dominating Set, Connected Dominating Set, and Total Dominating Set are NP-complete in circle graphs. To the best of our knowledge, nothing was known about the parameterized complexity of these problems in circle graphs. In this paper we prove the following results, which contribute in this direction: Dominating Set, Independent Dominating Set, Connected Dominating Set, Total Dominating Set, and Acyclic Dominating Set are W[1]-hard in circle graphs, parameterized by the size of the solution. Whereas both Connected Dominating Set and Acyclic Dominating Set are W[1]-hard in circle graphs, it turns out that Connected Acyclic Dominating Set is polynomial-time solvable in circle graphs. If T is a given tree, deciding whether a circle graph has a dominating set isomorphic to T is NP-complete when T is in the input, and FPT when parameterized by |V(T)|. We prove that the FPT algorithm is subexponential
Containment orders – a lifelong journey
AbstractMathematical explorations steer us through fields and forests of applications where science and computing meet discrete structures and combinatorics. Today’s excursion takes us on one of the speaker’s favorite journeys in the world of algorithmic graph theory – containment orders, treelike orders and their comparability graphs
- …