54 research outputs found
Subclasses of Normal Helly Circular-Arc Graphs
A Helly circular-arc model M = (C,A) is a circle C together with a Helly
family \A of arcs of C. If no arc is contained in any other, then M is a proper
Helly circular-arc model, if every arc has the same length, then M is a unit
Helly circular-arc model, and if there are no two arcs covering the circle,
then M is a normal Helly circular-arc model. A Helly (resp. proper Helly, unit
Helly, normal Helly) circular-arc graph is the intersection graph of the arcs
of a Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc model.
In this article we study these subclasses of Helly circular-arc graphs. We show
natural generalizations of several properties of (proper) interval graphs that
hold for some of these Helly circular-arc subclasses. Next, we describe
characterizations for the subclasses of Helly circular-arc graphs, including
forbidden induced subgraphs characterizations. These characterizations lead to
efficient algorithms for recognizing graphs within these classes. Finally, we
show how do these classes of graphs relate with straight and round digraphs.Comment: 39 pages, 13 figures. A previous version of the paper (entitled
Proper Helly Circular-Arc Graphs) appeared at WG'0
Boxicity and separation dimension
A family of permutations of the vertices of a hypergraph is
called 'pairwise suitable' for if, for every pair of disjoint edges in ,
there exists a permutation in in which all the vertices in one
edge precede those in the other. The cardinality of a smallest such family of
permutations for is called the 'separation dimension' of and is denoted
by . Equivalently, is the smallest natural number so that
the vertices of can be embedded in such that any two
disjoint edges of can be separated by a hyperplane normal to one of the
axes. We show that the separation dimension of a hypergraph is equal to the
'boxicity' of the line graph of . This connection helps us in borrowing
results and techniques from the extensive literature on boxicity to study the
concept of separation dimension.Comment: This is the full version of a paper by the same name submitted to
WG-2014. Some results proved in this paper are also present in
arXiv:1212.6756. arXiv admin note: substantial text overlap with
arXiv:1212.675
Boxicity of Line Graphs
Boxicity of a graph H, denoted by box(H), is the minimum integer k such that
H is an intersection graph of axis-parallel k-dimensional boxes in R^k. In this
paper, we show that for a line graph G of a multigraph, box(G) <=
2\Delta(\lceil log_2(log_2(\Delta)) \rceil + 3) + 1, where \Delta denotes the
maximum degree of G. Since \Delta <= 2(\chi - 1), for any line graph G with
chromatic number \chi, box(G) = O(\chi log_2(log_2(\chi))). For the
d-dimensional hypercube H_d, we prove that box(H_d) >= (\lceil log_2(log_2(d))
\rceil + 1)/2. The question of finding a non-trivial lower bound for box(H_d)
was left open by Chandran and Sivadasan in [L. Sunil Chandran and Naveen
Sivadasan. The cubicity of Hypercube Graphs. Discrete Mathematics,
308(23):5795-5800, 2008]. The above results are consequences of bounds that we
obtain for the boxicity of fully subdivided graphs (a graph which can be
obtained by subdividing every edge of a graph exactly once).Comment: 14 page
Extremal Values of the Interval Number of a Graph
The interval number of a simple graph is the smallest number such that to each vertex in there can be assigned a collection of at most finite closed intervals on the real line so that there is an edge between vertices and in if and only if some interval for intersects some interval for . The well known interval graphs are precisely those graphs with . We prove here that for any graph with maximum degree . This bound is attained by every regular graph of degree with no triangles, so is best possible. The degree bound is applied to show that for graphs on vertices and for graphs with edges
Bipartite powers of k-chordal graphs
Let k be an integer and k \geq 3. A graph G is k-chordal if G does not have
an induced cycle of length greater than k. From the definition it is clear that
3-chordal graphs are precisely the class of chordal graphs. Duchet proved that,
for every positive integer m, if G^m is chordal then so is G^{m+2}.
Brandst\"adt et al. in [Andreas Brandst\"adt, Van Bang Le, and Thomas Szymczak.
Duchet-type theorems for powers of HHD-free graphs. Discrete Mathematics,
177(1-3):9-16, 1997.] showed that if G^m is k-chordal, then so is G^{m+2}.
Powering a bipartite graph does not preserve its bipartitedness. In order to
preserve the bipartitedness of a bipartite graph while powering Chandran et al.
introduced the notion of bipartite powering. This notion was introduced to aid
their study of boxicity of chordal bipartite graphs. Given a bipartite graph G
and an odd positive integer m, we define the graph G^{[m]} to be a bipartite
graph with V(G^{[m]})=V(G) and E(G^{[m]})={(u,v) | u,v \in V(G), d_G(u,v) is
odd, and d_G(u,v) \leq m}. The graph G^{[m]} is called the m-th bipartite power
of G.
In this paper we show that, given a bipartite graph G, if G is k-chordal then
so is G^{[m]}, where k, m are positive integers such that k \geq 4 and m is
odd.Comment: 10 page
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