Volume 185, Issue 3

1. Perfect domination, Roman domination and perfect Roman domination in lexicographic product graphs

A. Cabrera Martinez ; C. Garcia-Gomez ; J. A. Rodriguez-Velazquez.
The aim of this paper is to obtain closed formulas for the perfect domination number, the Roman domination number and the perfect Roman domination number of lexicographic product graphs. We show that these formulas can be obtained relatively easily for the case of the first two parameters. The picture is quite different when it concerns the perfect Roman domination number. In this case, we obtain general bounds and then we give sufficient and/or necessary conditions for the bounds to be achieved. We also discuss the case of perfect Roman graphs and we characterize the lexicographic product graphs where the perfect Roman domination number equals the Roman domination number.

2. Coxeter invariants for non-negative unit forms of Dynkin type A

Jesús Arturo Jiménez González.
Two integral quadratic unit forms are called strongly Gram congruent if their upper triangular Gram matrices are Z-congruent. The paper gives a combinatorial strong Gram invariant for those unit forms that are non-negative of Dynkin type A, within the framework introduced in [Fundamenta Informaticae 184(1):49-82, 2021], and uses it to determine all corresponding Coxeter polynomials and (reduced) Coxeter numbers.

3. Nominal Unification and Matching of Higher Order Expressions with Recursive Let

Manfred Schmidt-Schauß ; Temur Kutsia ; Jordi Levy ; Mateu Villaret ; Yunus Kutz.
A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in nondeterministic polynomial time. We also explore specializations like nominal letrec-matching for expressions, for DAGs, and for garbage-free expressions and determine their complexity. We also provide a nominal unification algorithm for higher-order expressions with recursive let and atom-variables, where we show that it also runs in nondeterministic polynomial time. In addition we prove that there is a guessing strategy for nominal unification with letrec and atom-variable that is a trade-off between exponential growth and non-determinism. Nominal matching with variables representing partial letrec-environments is also shown to be in NP.