6,538 research outputs found
Computing maximum cliques in -EPG graphs
EPG graphs, introduced by Golumbic et al. in 2009, are edge-intersection
graphs of paths on an orthogonal grid. The class -EPG is the subclass of
EPG graphs where the path on the grid associated to each vertex has at most
bends. Epstein et al. showed in 2013 that computing a maximum clique in
-EPG graphs is polynomial. As remarked in [Heldt et al., 2014], when the
number of bends is at least , the class contains -interval graphs for
which computing a maximum clique is an NP-hard problem. The complexity status
of the Maximum Clique problem remains open for and -EPG graphs. In
this paper, we show that we can compute a maximum clique in polynomial time in
-EPG graphs given a representation of the graph.
Moreover, we show that a simple counting argument provides a
-approximation for the coloring problem on -EPG graphs without
knowing the representation of the graph. It generalizes a result of [Epstein et
al, 2013] on -EPG graphs (where the representation was needed)
Partitioning Perfect Graphs into Stars
The partition of graphs into "nice" subgraphs is a central algorithmic
problem with strong ties to matching theory. We study the partitioning of
undirected graphs into same-size stars, a problem known to be NP-complete even
for the case of stars on three vertices. We perform a thorough computational
complexity study of the problem on subclasses of perfect graphs and identify
several polynomial-time solvable cases, for example, on interval graphs and
bipartite permutation graphs, and also NP-complete cases, for example, on grid
graphs and chordal graphs.Comment: Manuscript accepted to Journal of Graph Theor
Framework for Clique-based Fusion of Graph Streams in Multi-function System Testing
The paper describes a framework for multi-function system testing.
Multi-function system testing is considered as fusion (or revelation) of
clique-like structures. The following sets are considered: (i) subsystems
(system parts or units / components / modules), (ii) system functions and a
subset of system components for each system function, and (iii) function
clusters (some groups of system functions which are used jointly). Test
procedures (as units testing) are used for each subsystem. The procedures lead
to an ordinal result (states, colors) for each component, e.g., [1,2,3,4]
(where 1 corresponds to 'out of service', 2 corresponds to 'major faults', 3
corresponds to 'minor faults', 4 corresponds to 'trouble free service'). Thus,
for each system function a graph over corresponding system components is
examined while taking into account ordinal estimates/colors of the components.
Further, an integrated graph (i.e., colored graph) for each function cluster is
considered (this graph integrates the graphs for corresponding system
functions). For the integrated graph (for each function cluster) structure
revelation problems are under examination (revelation of some subgraphs which
can lead to system faults): (1) revelation of clique and quasi-clique (by
vertices at level 1, 2, etc.; by edges/interconnection existence) and (2)
dynamical problems (when vertex colors are functions of time) are studied as
well: existence of a time interval when clique or quasi-clique can exist.
Numerical examples illustrate the approach and problems.Comment: 6 pages, 13 figure
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