25,802 research outputs found
Rectilinear Planarity of Partial 2-Trees
A graph is rectilinear planar if it admits a planar orthogonal drawing
without bends. While testing rectilinear planarity is NP-hard in general (Garg
and Tamassia, 2001), it is a long-standing open problem to establish a tight
upper bound on its complexity for partial 2-trees, i.e., graphs whose
biconnected components are series-parallel. We describe a new O(n^2)-time
algorithm to test rectilinear planarity of partial 2-trees, which improves over
the current best bound of O(n^3 \log n) (Di Giacomo et al., 2022). Moreover,
for partial 2-trees where no two parallel-components in a biconnected component
share a pole, we are able to achieve optimal O(n)-time complexity. Our
algorithms are based on an extensive study and a deeper understanding of the
notion of orthogonal spirality, introduced several years ago (Di Battista et
al, 1998) to describe how much an orthogonal drawing of a subgraph is rolled-up
in an orthogonal drawing of the graph.Comment: arXiv admin note: substantial text overlap with arXiv:2110.00548
Appears in the Proceedings of the 30th International Symposium on Graph
Drawing and Network Visualization (GD 2022
A simple linear time algorithm for the locally connected spanning tree problem on maximal planar chordal graphs
A locally connected spanning tree (LCST) T of a graph G is a spanning tree of G such that, for each node, its neighborhood in T induces a connected sub- graph in G. The problem of determining whether a graph contains an LCST or not has been proved to be NP-complete, even if the graph is planar or chordal. The main result of this paper is a simple linear time algorithm that, given a maximal planar chordal graph, determines in linear time whether it contains an LCST or not, and produces one if it exists. We give an anal- ogous result for the case when the input graph is a maximal outerplanar graph
Operads and Phylogenetic Trees
We construct an operad whose operations are the edge-labelled
trees used in phylogenetics. This operad is the coproduct of ,
the operad for commutative semigroups, and , the operad with unary
operations corresponding to nonnegative real numbers, where composition is
addition. We show that there is a homeomorphism between the space of -ary
operations of and , where
is the space of metric -trees introduced by Billera, Holmes
and Vogtmann. Furthermore, we show that the Markov models used to reconstruct
phylogenetic trees from genome data give coalgebras of . These
always extend to coalgebras of the larger operad ,
since Markov processes on finite sets converge to an equilibrium as time
approaches infinity. We show that for any operad , its coproduct with
contains the operad constucted by Boardman and Vogt. To
prove these results, we explicitly describe the coproduct of operads in terms
of labelled trees.Comment: 48 pages, 3 figure
Partial magmatic bialgebras
A partial magmatic bialgebra, (T;S)-magmatic bialgebra where T \subset S are
subsets of the set of positive integers, is a vector space endowed with an
n-ary operation for each n in S and an m-ary co-operation for each m in T
satisfying some compatibility and unitary relations. We prove an analogue of
the Poincar\'e-Birkhoff-Witt theorem for these partial magmatic bialgebras.Comment: Revised version, after suggestions of the anonymous referee, 20 page
Long induced paths in graphs
We prove that every 3-connected planar graph on vertices contains an
induced path on vertices, which is best possible and improves
the best known lower bound by a multiplicative factor of . We
deduce that any planar graph (or more generally, any graph embeddable on a
fixed surface) with a path on vertices, also contains an induced path on
vertices. We conjecture that for any , there is a
contant such that any -degenerate graph with a path on vertices
also contains an induced path on vertices. We provide
examples showing that this order of magnitude would be best possible (already
for chordal graphs), and prove the conjecture in the case of interval graphs.Comment: 20 pages, 5 figures - revised versio
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