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Antonios Kalampakas

Publications and source records attributed to Antonios Kalampakas.

9 recordsLinked to original sources
arXiv2026

Finite tests and low-rank excluded minors for matroid path-width two

The path-width of a matroid is the minimum, over all orderings of its ground set, of the maximum value of its connectivity function on an initial segment. For any two connected rooted matroids whose roots are neither loops nor coloops, we form the 2-sum of each with a third such rooted matroid by identifying their roots and deleting the common element. The two original rooted matroids are equivalent when these sums agree on whether their path-width is at most two for every choice of the third rooted matroid. We prove that the two rooted matroids are equivalent if and only if this agreement holds for thirteen fixed graphic or cographic rooted matroids. They distinguish exactly twenty equivalence classes, and, moreover, there exists no family of twelve fixed connected rooted matroids that can guarantee this. Using these classes, we show that the tree in the canonical tree decomposition of an excluded minor for path-width at most two has diameter at most ten edges. For a connected matroid given by a tree decomposition using 2-sums, we give an exact recurrence to decide whether its path-width is at most two and to construct an ordering of width at most two when possible. Moreover, we derive a formula for the path-width of parallel extensions that gives complete binary excluded-minor lists in ranks three, four and five, containing two, four and eleven isomorphism classes, respectively.

math.CO↗
arXiv2026

Decomposition Profiles and Weisfeiler-Leman Dimension for Graphs of Bounded Rank Width

The Weisfeiler-Leman (WL) algorithm tests graph isomorphism by assigning colours to $d$-tuples of vertices and iteratively refining the colouring according to neighbourhood configurations until it stabilizes. Increasing $d$ allows the test to detect finer structural differences between graphs, but the computational cost grows exponentially with $d$. The central question is which dimension allows us to distinguish nonisomorphic graphs. We relate a sufficient such dimension to the structure of a graph $G$ through a rank decomposition over $\mathbb F_2$. The two parameters governing the bound are the maximum cut rank $w$ and the fork load $\ell$, which is the largest sum of the parent and two child cut ranks at a fork in the decomposition tree. We prove that WL in dimension $\max\{\ell+1,2w+2\}$ distinguishes every finite nonempty vertex-coloured graph $G$ with such a decomposition from every graph not isomorphic to $G$. Consequently, for $k\geq1$, graphs of rank width at most $k$ can be distinguished from every nonisomorphic graph in dimension $3k+1$, graphs of linear rank width at most $k$ in dimension $2k+2$, and edgeless graphs in dimension one. Uncoloured Cai-Fürer-Immerman (CFI) constructions give linear lower bounds for both width parameters, so the upper bounds are tight up to constant factors.

math.CO↗
arXiv2026

Graphoid Automata, Frobenius PROPs, and Semiring Evaluation

Graphoid automata evaluate graphs by composing relations between words of states. We examine the equations that make this evaluation independent of the chosen graph expression, and what changes when relations are replaced by matrices over a commutative semiring. We prove that the classical graphoid equations are equivalent to the special commutative Frobenius equations together with the required symmetry laws. The known cospan and hypergraph presentations then describe equality of structural and labelled expressions, with the bone law deleting precisely the isolated internal vertices. We use established partition representations to compare this equality with equality of evaluated matrices. For any nonempty finite state set $Q$ and commutative semiring $K$, we give a uniform criterion for two structural expressions to have the same matrix. The criterion determines which isolated-vertex counts are distinguished and when a vanishing scalar makes matrices from different boundary partitions equal. For the canonical structure, the bone law holds precisely when $|Q|\cdot1_K=1_K$. Preserving all relational compositions is a separate requirement, equivalent to additive idempotency for $|Q|\geq2$. We then examine the characteristic matrices of the known relational structures on $Q$ given by disjoint unions of Abelian groups. Their Frobenius equations hold over every commutative semiring, whereas specialness depends on the order of each group and the bone law depends on the number of groups. These conditions specify which equations remain valid when weights are assigned to a graphoid automaton.

math.CT↗
arXiv2026

Recognizability equals CMSO-definability for graphs of rank-width at most two

We prove that, on finite graphs of rank-width at most two, VR-recognizability and counting monadic second-order definability coincide. This advances the recognizability-versus-definability problem from bounded linear clique-width to the first nontrivial bounded rank-width level beyond the rank-width-one split-decomposition case. The proof first treats split-prime graphs. The maximal partial-tree theory of Clark and Whittle organizes the non-sequential cut-rank-two separations, while a single strong separation orients all strong equivalence classes and yields a CMSO-definable laminar family of canonical cores. Although the auxiliary partial tree is not itself transduced, it proves that every canonical local piece has a port-contiguous layout of uniformly bounded linear rank-width. The width argument uses partition atoms and the branch-width-three display theorem of Hall, Oxley, Semple, and Whittle and does not assume that graph torsos remain prime. Coherent ordered rank-two frames then permit a finite-state bottom-up evaluation whose local transitions are definable by the bounded-linear-clique-width theorem of Bojańczyk, Grohe, and Pilipczuk. Finally, the CMSO-transducible canonical split decomposition lifts the result from prime graphs to arbitrary graphs of rank-width at most two.

math.CO↗
arXiv2026

Width-Bounded Equational Derivations for Finite Graph Expressions

Completeness of an equational presentation guarantees an equality path but need not control the resources used along it. For finite graph expressions we measure derivational space by the largest input-output interface of an intermediate raw term. For every finite doubly ranked edge alphabet $Σ$, we prove that equal closed expressions of pattern width at most $k$ are joined by a derivation in which every step applies an equation in either direction and every intermediate width is at most a computable $B_Σ(k)$, independently of graph size. The derivation uses only the structural magmoid laws and the fifteen finite-graphoid schemes. The construction compiles each expression through protected cores and finite routing windows to an encoding-canonical representative $\operatorname{NF}_k$ of width at most $4k+4$. We call this property bounded equational coherence. We also prove $\operatorname{bw}(F)\le\operatorname{patw}(F)\le4(\operatorname{tw}(F)+1)$ whenever the underlying simple graph has at least two edges, separate layered linear from branching expressions on cliques, and provide machine-checkable witnesses and finite invariants for the first five nontrivial clique values.

cs.LO↗
arXiv2026

Automatic constraints with few subpowers and graphoid recognition

Finite automata can describe relations of unbounded arity that are exponentially larger than their descriptions. We prove that constraint satisfaction for such relations is solvable in polynomial time whenever their length slices are preserved by a common fixed edge operation on a finite domain. The algorithm computes compact representations of the complete solution relation and its projections. Its main ingredient is a polynomial-time compilation of nondeterministic finite automata into the fork witnesses and small projections required by the few-subpowers algorithm. In the Mal'tsev case, a direct proof is polynomial also when the domain and operation table are supplied as input, answering the Mal'tsev tractability question for automatic constraint satisfaction. We also characterize all invariant relations of a family of 3-edge algebras with neither Mal'tsev nor near-unanimity terms. Their normal forms combine Boolean activity constraints with affine value spaces and yield canonical quadratic-bit representations constructible from NFAs or arbitrary generators. For graphoid automata, these results give polynomial-time recognition without a graph-width restriction, effective boundary composition, and comparison of finite graph relations. The quadratic boundary bounds are optimal in the worst case. A fixed three-state example separates polynomial-time recognition from hard exact counting.

cs.LO↗
arXiv2026

Split-Free Cable Expressions: Active Neighborhood Profiles and Linear Rank-Width

We introduce split-free cable expressions and their sequential restriction. Live cables are vertex blocks that future operations cannot split. The main result identifies sequential split-free cable width, up to an additive two, with active neighborhood-width, the minimum number of distinct nonzero future-neighborhood classes across a linear layout. The lower bound extracts such a layout from every cable play and includes a parity argument for prefixes inside a birth group. The upper bound compiles any active-neighborhood layout into a singleton-birth play. This gives an exponential upper bound and a linear lower bound in terms of linear rank-width. We prove that the complement of the seven-vertex cycle has linear rank-width two and sequential cable width exactly eight. This disproves the proposed affine upper estimate already at rank two. We also prove an unbounded separation between branching expression width and sequential width on trees.

math.CO↗
arXiv2024

The Price of Cognition and Replicator Equations in Parallel Neural Networks

In this paper, we are aiming to propose a novel mathematical model that studies the dynamics of synaptic damage in terms of concentrations of toxic neuropeptides/neurotransmitters during neurotransmission processes. Our primary objective is to employ Wardrop's first and second principles within a neural network of the brain. In order to comprehensively incorporate Wardrop's first and second principles into the neural network of the brain, we introduce two novel concepts: \textit{neuropeptide's (neurotransmitter's) equilibrium} and \textit{synapses optimum}. The \textit{neuropeptide/neurotransmitter equilibrium} refers to \textit{a distribution of toxic neuropeptides/neurotransmitters that leads to uniform damage across all synaptic links}. Meanwhile, \textit{synapses optimum} is \textit{the most desirable distribution of toxic neuropeptides/neurotransmitters that minimizes the cumulative damage experienced by all synapses}. In the context of a neural network within the brain, an analogue of the price of anarchy is \textit{the price of cognition} which is \textit{the most unfavorable ratio between the overall impairment caused by toxic neuropeptide's (neurotransmitter's) equilibrium in comparison to the optimal state of synapses (synapses optimum)}. To put it differently, \textit{the price of cognition} measures \textit{the loss of cognitive ability resulting from increased concentrations of toxic neuropeptides/neurotransmitters}. Additionally, a replicator equation is proposed within this framework that leads to the establishment of the synapses optimum during the neurotransmission process.

q-bio.NC↗