115 research outputs found
Boxicity and Cubicity of Asteroidal Triple free graphs
An axis parallel -dimensional box is the Cartesian product where each is a closed interval on the real line.
The {\it boxicity} of a graph , denoted as \boxi(G), is the minimum
integer such that can be represented as the intersection graph of a
collection of -dimensional boxes. An axis parallel unit cube in
-dimensional space or a -cube is defined as the Cartesian product where each is a closed interval on the
real line of the form . The {\it cubicity} of , denoted as
\cub(G), is the minimum integer such that can be represented as the
intersection graph of a collection of -cubes.
Let denote a star graph on nodes. We define {\it claw number} of
a graph as the largest positive integer such that is an induced
subgraph of and denote it as \claw.
Let be an AT-free graph with chromatic number and claw number
\claw. In this paper we will show that \boxi(G) \leq \chi(G) and this bound
is tight. We also show that \cub(G) \leq \boxi(G)(\ceil{\log_2 \claw} +2)
\chi(G)(\ceil{\log_2 \claw} +2). If is an AT-free graph having
girth at least 5 then \boxi(G) \leq 2 and therefore \cub(G) \leq
2\ceil{\log_2 \claw} +4.Comment: 15 pages: We are replacing our earlier paper regarding boxicity of
permutation graphs with a superior result. Here we consider the boxicity of
AT-free graphs, which is a super class of permutation graph
Open problems on graph coloring for special graph classes.
For a given graph G and integer k, the Coloring problem is that of testing whether G has a k-coloring, that is, whether there exists a vertex mapping c:V→{1,2,…}c:V→{1,2,…} such that c(u)≠c(v)c(u)≠c(v) for every edge uv∈Euv∈E. We survey known results on the computational complexity of Coloring for graph classes that are hereditary or for which some graph parameter is bounded. We also consider coloring variants, such as precoloring extensions and list colorings and give some open problems in the area of on-line coloring
Obstructions to Faster Diameter Computation: Asteroidal Sets
Full version of an IPEC'22 paperAn extremity is a vertex such that the removal of its closed neighbourhood does not increase the number of connected components. Let be the class of all connected graphs whose quotient graph obtained from modular decomposition contains no more than pairwise nonadjacent extremities. Our main contributions are as follows. First, we prove that the diameter of every -edge graph in can be computed in deterministic time. We then improve the runtime to linear for all graphs with bounded clique-number. Furthermore, we can compute an additive -approximation of all vertex eccentricities in deterministic time. This is in sharp contrast with general -edge graphs for which, under the Strong Exponential Time Hypothesis (SETH), one cannot compute the diameter in time for any . As important special cases of our main result, we derive an -time algorithm for exact diameter computation within dominating pair graphs of diameter at least six, and an -time algorithm for this problem on graphs of asteroidal number at most . We end up presenting an improved algorithm for chordal graphs of bounded asteroidal number, and a partial extension of our results to the larger class of all graphs with a dominating target of bounded cardinality. Our time upper bounds in the paper are shown to be essentially optimal under plausible complexity assumptions
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