110,977 research outputs found
On Fork-free T-perfect Graphs
In an attempt to understanding the complexity of the independent set problem,
Chv{\'a}tal defined t-perfect graphs. While a full characterization of this
class is still at large, progress has been achieved for claw-free graphs [Bruhn
and Stein, Math.\ Program.\ 2012] and -free graphs [Bruhn and Fuchs,
SIAM J.\ Discrete Math.\ 2017]. We take one more step to characterize fork-free
t-perfect graphs, and show that they are strongly t-perfect and
three-colorable. We also present polynomial-time algorithms for recognizing and
coloring these graphs
A semi-induced subgraph characterization of upper domination perfect graphs
Let β(G) and Γ(G) be the independence number and the upper domination number of a graph G, respectively. A graph G is called Γ-perfect if β(H) = Γ(H), for every induced subgraph H of G. The class of Γ-perfect graphs generalizes such well-known classes of graphs as strongly perfect graphs, absorbantly perfect graphs, and circular arc graphs. In this article, we present a characterization of Γ-perfect graphs in terms of forbidden semi-induced subgraphs. Key roles in the characterization are played by the odd prism and the even Möbius ladder, where the prism and the Möbius ladder are well-known 3-regular graphs [2]. Using the semi-induced subgraph characterization, we obtain a characterization of K 1.3-free Γ-perfect graphs in terms of forbidden induced subgraphs. © 1999 John Wiley & Sons, Inc
Stimulated Raman adiabatic passage-like protocols for amplitude transfer generalize to many bipartite graphs
Adiabatic passage techniques, used to drive a system from one quantum state
into another, find widespread application in physics and chemistry. We focus on
techniques to spatially transport a quantum amplitude over a strongly coupled
system, such as STImulated Raman Adiabatic Passage (STIRAP) and Coherent
Tunnelling by Adiabatic Passage (CTAP). Previous results were shown to work on
certain graphs, such as linear chains, square and triangular lattices, and
branched chains. We prove that similar protocols work much more generally, in a
large class of (semi-)bipartite graphs. In particular, under random couplings,
adiabatic transfer is possible on graphs that admit a perfect matching both
when the sender is removed and when the receiver is removed. Many of the
favorable stability properties of STIRAP/CTAP are inherited, and our results
readily apply to transfer between multiple potential senders and receivers. We
numerically test transfer between the leaves of a tree, and find surprisingly
accurate transfer, especially when straddling is used. Our results may find
applications in short-distance communication between multiple quantum
computers, and open up a new question in graph theory about the spectral gap
around the value 0.Comment: 17 pages, 3 figures. v2 is made more mathematical and precise than v
Clique-Stable Set separation in perfect graphs with no balanced skew-partitions
Inspired by a question of Yannakakis on the Vertex Packing polytope of
perfect graphs, we study the Clique-Stable Set Separation in a non-hereditary
subclass of perfect graphs. A cut (B,W) of G (a bipartition of V(G)) separates
a clique K and a stable set S if and . A
Clique-Stable Set Separator is a family of cuts such that for every clique K,
and for every stable set S disjoint from K, there exists a cut in the family
that separates K and S. Given a class of graphs, the question is to know
whether every graph of the class admits a Clique-Stable Set Separator
containing only polynomially many cuts. It is open for the class of all graphs,
and also for perfect graphs, which was Yannakakis' original question. Here we
investigate on perfect graphs with no balanced skew-partition; the balanced
skew-partition was introduced in the proof of the Strong Perfect Graph Theorem.
Recently, Chudnovsky, Trotignon, Trunck and Vuskovic proved that forbidding
this unfriendly decomposition permits to recursively decompose Berge graphs
using 2-join and complement 2-join until reaching a basic graph, and they found
an efficient combinatorial algorithm to color those graphs. We apply their
decomposition result to prove that perfect graphs with no balanced
skew-partition admit a quadratic-size Clique-Stable Set Separator, by taking
advantage of the good behavior of 2-join with respect to this property. We then
generalize this result and prove that the Strong Erdos-Hajnal property holds in
this class, which means that every such graph has a linear-size biclique or
complement biclique. This property does not hold for all perfect graphs (Fox
2006), and moreover when the Strong Erdos-Hajnal property holds in a hereditary
class of graphs, then both the Erdos-Hajnal property and the polynomial
Clique-Stable Set Separation hold.Comment: arXiv admin note: text overlap with arXiv:1308.644
The chromatic index of strongly regular graphs
We determine (partly by computer search) the chromatic index (edge-chromatic
number) of many strongly regular graphs (SRGs), including the SRGs of degree and their complements, the Latin square graphs and their complements,
and the triangular graphs and their complements. Moreover, using a recent
result of Ferber and Jain it is shown that an SRG of even order , which is
not the block graph of a Steiner 2-design or its complement, has chromatic
index , when is big enough. Except for the Petersen graph, all
investigated connected SRGs of even order have chromatic index equal to their
degree, i.e., they are class 1, and we conjecture that this is the case for all
connected SRGs of even order.Comment: 10 page
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