102 research outputs found
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
Characterizing 2-crossing-critical graphs
It is very well-known that there are precisely two minimal non-planar graphs:
and (degree 2 vertices being irrelevant in this context). In
the language of crossing numbers, these are the only 1-crossing-critical
graphs: they each have crossing number at least one, and every proper subgraph
has crossing number less than one. In 1987, Kochol exhibited an infinite family
of 3-connected, simple 2-crossing-critical graphs. In this work, we: (i)
determine all the 3-connected 2-crossing-critical graphs that contain a
subdivision of the M\"obius Ladder ; (ii) show how to obtain all the
not 3-connected 2-crossing-critical graphs from the 3-connected ones; (iii)
show that there are only finitely many 3-connected 2-crossing-critical graphs
not containing a subdivision of ; and (iv) determine all the
3-connected 2-crossing-critical graphs that do not contain a subdivision of
.Comment: 176 pages, 28 figure
Towards obtaining a 3-Decomposition from a perfect Matching
A decomposition of a graph is a set of subgraphs whose edges partition those
of . The 3-decomposition conjecture posed by Hoffmann-Ostenhof in 2011
states that every connected cubic graph can be decomposed into a spanning tree,
a 2-regular subgraph, and a matching. It has been settled for special classes
of graphs, one of the first results being for Hamiltonian graphs. In the past
two years several new results have been obtained, adding the classes of plane,
claw-free, and 3-connected tree-width 3 graphs to the list.
In this paper, we regard a natural extension of Hamiltonian graphs: removing
a Hamiltonian cycle from a cubic graph leaves a perfect matching. Conversely,
removing a perfect matching from a cubic graph leaves a disjoint union
of cycles. Contracting these cycles yields a new graph . The graph is
star-like if is a star for some perfect matching , making Hamiltonian
graphs star-like. We extend the technique used to prove that Hamiltonian graphs
satisfy the 3-decomposition conjecture to show that 3-connected star-like
graphs satisfy it as well.Comment: 21 pages, 7 figure
On the purity of minor-closed classes of graphs
Given a graph H with at least one edge, let gapH(n) denote the maximum difference between the numbers of edges in two n-vertex edge-maximal graphs with no minor H. We show that for exactly four connected graphs H (with at least two vertices), the class of graphs with no minor H is pure, that is, gapH(n) = 0 for all n ≥ 1; and for each connected graph H (with at least two vertices) we have the dichotomy that either gapH(n) = O(1) or gapH(n) = ⊝(n). Further, if H is 2-connected and does not yield a pure class, then there is a constant c > 0 such that gapH(n) ∼ cn. We also give some partial results when H is not connected or when there are two or more excluded minors
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