825 research outputs found
On graphs with no induced subdivision of
We prove a decomposition theorem for graphs that do not contain a subdivision
of as an induced subgraph where is the complete graph on four
vertices. We obtain also a structure theorem for the class of graphs
that contain neither a subdivision of nor a wheel as an induced subgraph,
where a wheel is a cycle on at least four vertices together with a vertex that
has at least three neighbors on the cycle. Our structure theorem is used to
prove that every graph in is 3-colorable and entails a polynomial-time
recognition algorithm for membership in . As an intermediate result, we
prove a structure theorem for the graphs whose cycles are all chordless
On graphs that do not contain a subdivision of the complete graph on four vertices as an induced subgraph
We prove a decomposition theorem for graphs that do not contain a subdivision of the complete graph on four vertices as an induced subgraph.Induced, subgraph, decomposition.
Span programs and quantum algorithms for st-connectivity and claw detection
We introduce a span program that decides st-connectivity, and generalize the
span program to develop quantum algorithms for several graph problems. First,
we give an algorithm for st-connectivity that uses O(n d^{1/2}) quantum queries
to the n x n adjacency matrix to decide if vertices s and t are connected,
under the promise that they either are connected by a path of length at most d,
or are disconnected. We also show that if T is a path, a star with two
subdivided legs, or a subdivision of a claw, its presence as a subgraph in the
input graph G can be detected with O(n) quantum queries to the adjacency
matrix. Under the promise that G either contains T as a subgraph or does not
contain T as a minor, we give O(n)-query quantum algorithms for detecting T
either a triangle or a subdivision of a star. All these algorithms can be
implemented time efficiently and, except for the triangle-detection algorithm,
in logarithmic space. One of the main techniques is to modify the
st-connectivity span program to drop along the way "breadcrumbs," which must be
retrieved before the path from s is allowed to enter t.Comment: 18 pages, 4 figure
Finding an induced subdivision of a digraph
We consider the following problem for oriented graphs and digraphs: Given an
oriented graph (digraph) , does it contain an induced subdivision of a
prescribed digraph ? The complexity of this problem depends on and on
whether must be an oriented graph or is allowed to contain 2-cycles. We
give a number of examples of polynomial instances as well as several
NP-completeness proofs
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