2,127 research outputs found
Uniqueness and minimal obstructions for tree-depth
A k-ranking of a graph G is a labeling of the vertices of G with values from
{1,...,k} such that any path joining two vertices with the same label contains
a vertex having a higher label. The tree-depth of G is the smallest value of k
for which a k-ranking of G exists. The graph G is k-critical if it has
tree-depth k and every proper minor of G has smaller tree-depth.
We establish partial results in support of two conjectures about the order
and maximum degree of k-critical graphs. As part of these results, we define a
graph G to be 1-unique if for every vertex v in G, there exists an optimal
ranking of G in which v is the unique vertex with label 1. We show that several
classes of k-critical graphs are 1-unique, and we conjecture that the property
holds for all k-critical graphs. Generalizing a previously known construction
for trees, we exhibit an inductive construction that uses 1-unique k-critical
graphs to generate large classes of critical graphs having a given tree-depth.Comment: 14 pages, 4 figure
An FPT algorithm and a polynomial kernel for Linear Rankwidth-1 Vertex Deletion
Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and
Seymour [Approximating clique-width and branch-width. J. Combin. Theory Ser. B,
96(4):514--528, 2006]. Motivated from recent development on graph modification
problems regarding classes of graphs of bounded treewidth or pathwidth, we
study the Linear Rankwidth-1 Vertex Deletion problem (shortly, LRW1-Vertex
Deletion). In the LRW1-Vertex Deletion problem, given an -vertex graph
and a positive integer , we want to decide whether there is a set of at most
vertices whose removal turns into a graph of linear rankwidth at most
and find such a vertex set if one exists. While the meta-theorem of
Courcelle, Makowsky, and Rotics implies that LRW1-Vertex Deletion can be solved
in time for some function , it is not clear whether this
problem allows a running time with a modest exponential function.
We first establish that LRW1-Vertex Deletion can be solved in time . The major obstacle to this end is how to handle a long
induced cycle as an obstruction. To fix this issue, we define necklace graphs
and investigate their structural properties. Later, we reduce the polynomial
factor by refining the trivial branching step based on a cliquewidth expression
of a graph, and obtain an algorithm that runs in time . We also prove that the running time cannot be improved to under the Exponential Time Hypothesis assumption. Lastly,
we show that the LRW1-Vertex Deletion problem admits a polynomial kernel.Comment: 29 pages, 9 figures, An extended abstract appeared in IPEC201
Matrix partitions of perfect graphs
AbstractGiven a symmetric m by m matrix M over 0,1,*, the M-partition problem asks whether or not an input graph G can be partitioned into m parts corresponding to the rows (and columns) of M so that two distinct vertices from parts i and j (possibly with i=j) are non-adjacent if M(i,j)=0, and adjacent if M(i,j)=1. These matrix partition problems generalize graph colourings and homomorphisms, and arise frequently in the study of perfect graphs; example problems include split graphs, clique and skew cutsets, homogeneous sets, and joins.In this paper we study M-partitions restricted to perfect graphs. We identify a natural class of ‘normal’ matrices M for which M-partitionability of perfect graphs can be characterized by a finite family of forbidden induced subgraphs (and hence admits polynomial time algorithms for perfect graphs). We further classify normal matrices into two classes: for the first class, the size of the forbidden subgraphs is linear in the size of M; for the second class we only prove exponential bounds on the size of forbidden subgraphs. (We exhibit normal matrices of the second class for which linear bounds do not hold.)We present evidence that matrices M which are not normal yield badly behaved M-partition problems: there are polynomial time solvable M-partition problems that do not have finite forbidden subgraph characterizations for perfect graphs. There are M-partition problems that are NP-complete for perfect graphs. There are classes of matrices M for which even proving ‘dichotomy’ of the corresponding M-partition problems for perfect graphs—i.e., proving that these problems are all polynomial or NP-complete—is likely to be difficult
Higher-Spin Gauge Theories and Bulk Locality
We present a no-go result on consistent Noether interactions among
higher-spin gauge fields on anti-de Sitter space-times. We show that there is a
non-local obstruction at the classical level to consistent interacting field
theory descriptions of massless higher-spin particles that are described in the
free limit by the free Fronsdal action, under the assumption that such theories
arise from the gauging of a global higher-spin symmetry. Our result suggests
that the Fronsdal programme for introducing interactions among higher-spin
gauge fields cannot be completed without introducing new guiding principles,
which could lie beyond the framework of classical field theory.Comment: 10 pages + refs. v2: Improved presentation, clarifications and
references added. Some parts removed which just reviewed previous published
work. To appear in Physical Review Letter
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