Abstract

The competition graph and its generalizations have interested graph theorists for over 30 years. Recently, Cho, Kim, and Nam introduced the m-step competition graph and computed the 2-step competition numbers of paths and cycles. We extend their results in a partial determination of the m-step competition numbers of paths and cycles. In addition, we introduce two new variants of competition graphs: same-step and any-step. We classify same-step and any-step competition graphs and investigate their related competition numbers.

Highlights

  • Over forty years have passed since the birth of the Constituent Quark Model, yet the scalar mesons still challenge theoreticians and experimentalists

  • An important problem is the understanding of the low energy part of the S-wave K −π + spectrum, where the existence of an I = 1/2 state, the κ(800) meson, has been the subject of a long-standing debate

  • Dalitz plot, applying the Model-Independent Partial Wave Analysis (MIPWA) technique, developed by the E791 Collaboration [12], to the same data set used for the K-matrix and Isobar fits [3]

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Summary

Introduction

Over forty years have passed since the birth of the Constituent Quark Model, yet the scalar mesons still challenge theoreticians and experimentalists. The BaBar Collaboration [6] uses the I = 1/2 elastic scattering amplitude from LASS for the K π S-wave Another approach is the non-parametric analysis of the K π amplitude from the D+ → K − K +π + decay performed by FOCUS [10]. Dalitz plot, applying the MIPWA technique, developed by the E791 Collaboration [12], to the same data set used for the K-matrix and Isobar fits [3]. The measured phase, in addition to the I = 1/2 K −π + phase, may contain contributions from the I = 3/2 components, as well as possible contributions from three-body final state interactions. The FOCUS spectrometer has a system of three multi-cell threshold Cerenkov counters to perform the charged particle identification, separating kaons from pions up to a momentum of 60 GeV/c. It is this interference with the P-wave that allows one to access the S-wave phase

The model independent partial-wave analysis formalism
Results of the MIPWA
Goodness-of-fit
Systematic uncertainties
Summary and conclusions

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