Asian Journal of Mathematics and Computer Research

Graph Embeddings on Surfaces: A Classical Review of Topological Graph Theory

Ghadeer Khudhair Obayes, Karrar Khudhair Obayes — Fri, 11 Sep 2026 00:00:00 +0000

Topological graph theory studies graphs in relation to the surfaces on which they can be drawn. This review presents the main classical ideas of the field in a clear and connected framework. It begins with graph embeddings, Euler’s formula, and the basic principles of planar graphs. It then discusses important classical results, including the theorems of Kuratowski, Whitney, Mac Lane, and Wagner, and explains their role in understanding planarity and graph structure. The review also introduces graph embeddings on more general surfaces and examines key concepts such as genus, rotation systems, dual graphs, and surface colouring. Further topics include maximum genus, genus distributions, excluded minors, and algorithms for deciding whether a graph can be embedded on a fixed surface. Particular attention is given to the relationships among these concepts. Euler-type arguments provide useful numerical restrictions, while subdivisions and graph minors give structural characterisations of embeddability. Rotation systems offer a combinatorial way to describe embeddings, whereas genus-related parameters measure different aspects of their topological complexity. Overall, the review provides an accessible account of the classical foundations of topological graph theory and shows how its major results form a unified framework for studying graphs on surfaces. The historical development of the Heawood map-colouring programme and its relation to complete-graph embeddings is treated explicitly, including the Klein bottle exception and the sphere as the Four Colour Theorem case. Recent work is used to show how classical Kuratowski-type and bounded-genus questions remain active. The final synthesis separates established results from continuing directions in obstruction enumeration, representativity, embedding distributions, and near-planar graph drawing.

Statistical Intelligence for Detecting AI Model Aging and Concept Drift in Adaptive Cybersecurity Systems

Sat, 05 Sep 2026 00:00:00 +0000

Machine-learning intrusion detectors are typically validated once at deployment and then trusted indefinitely. This is unsafe because the traffic distribution used for training differs from what deployed detectors encounter, with the gap widening as adversaries introduce unseen techniques. We formalise this degradation as model aging and develop a statistical decision framework that identifies, without ground-truth labels, when a cybersecurity model becomes unreliable and when retraining is economically justified.

The framework combines the Population Stability Index, Kullback–Leibler and Jensen–Shannon divergences, Wasserstein-1 distance, the two-sample Kolmogorov–Smirnov statistic, and sequential ADWIN, Page–Hinkley, and CUSUM procedures with two bounded state variables: the Model Aging Index (MAI), which accumulates null-calibrated distributional stress under exponential forgetting, and the Cybersecurity Drift Index (CDI), which incorporates the asymmetric costs of missed intrusions and false alerts. We prove that both indices are bounded, monotone, and Lipschitz, while their stationary exceedance probability admits an explicit bound for certified false-alarm control. The underlying marginal transport statistic also lower-bounds the joint Wasserstein radius, establishing alarm soundness.

Retraining is formulated as an optimal-stopping problem, yielding a control-limit policy analogous to an inventory replenishment threshold. Experiments on NSL-KDD introduce 17 genuinely novel attack types across 100 batches, five drift regimes, and seven architectures. Novel-family drift reduces attack recall from 0.99 to 0.57–0.66. The CDI controller cuts expected cost by 44.3% versus no retraining and 17.3% versus periodic retraining, while closing 79% of the oracle gap. Notably, periodic retraining can be harmful under adversarial drift.