Aida Abiad

Aida Abiad, Gabriel Coutinho, Emanuel Juliano et al.

In 1981, Lubiw proved that the fixed point free automorphism problem (FPFAut) is NP-complete: given a graph G, determine whether there exists an automorphism that maps no vertex of G to itself. We revisit this problem and prove that FPFAut remains NP-complete when restricted to split, bipartite, k-subdivided, and H-free graphs, if H is not an induced subgraph of P_4. The class of P_4-free graphs receives the special name of cographs. We provide a polynomial time algorithm for three extensions of cographs: bounded modular-width graphs, tree-cographs and P_4-sparse graphs. Our approach uses the well known modular decomposition of graphs. As a consequence, we generalize a result of Abiad et. al. on the problem of computing 2-homogeneous equitable partitions.

Aida Abiad, Antonina P. Khramova, Sven C. Polak et al.

The sum-rank metric provides a unifying framework that generalizes both the celebrated Hamming and rank metrics, and has found applications in areas such as network coding, distributed storage, and space-time coding. A central problem is to determine the maximum size of a code with prescribed minimum distance. In this paper, we derive new sharp upper bounds on the size of a sum-rank-metric code using spectral and optimization techniques, including a semidefinite programming (SDP) bound that can outperform the best existing bounds based on computational experiments. Furthermore, we compare the Delsarte linear programming (LP) bound and a recent eigenvalue LP bound, and show equivalences between them, with particular emphasis on extremal regimes of the sum-rank metric. Finally, we show how to use the several SDP, LP and eigenvalue bounds to prove non-existence results for certain optimal and perfect sum-rank metric codes. Our results suggest that the combination of spectral and optimization methods effectively captures the hybrid nature of the sum-rank metric, providing new techniques that overcome the limitations of classical coding-theoretic approaches.