We show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the underlying treewidth of a graph class $\mathcal{G}$ to be the minimum non-negative integer $c$ such that, for some function $f$, for every graph $G \in \mathcal{G}$ there is a graph $H$ with $\textrm{tw}(H) \leqslant c$ such that $G$ is isomorphic to a subgraph of $H \boxtimes K_{f(\textrm{tw}(G))}$. We introduce disjointed coverings of graphs and show they determine the underlying treewidth of any graph class. Using this result, we prove that the class of planar graphs has underlying treewidth $3$; the class of $K_{s,t}$-minor-free graphs has underlying treewidth $s$ (for $t \geqslant \max \{s,3\}$); and the class of $K_t$-minor-free graphs has underlying treewidth $t-2$. In general, we prove that a monotone class has bounded underlying treewidth if and only if it excludes some fixed topological minor. We also study the underlying treewidth of graph classes defined by an excluded subgraph or excluded induced subgraph. We show that the class of graphs with no $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a subdivided star, and that the class of graphs with no induced $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a star.

In France, the 2021 Act regarding sexual offences recognized incest as an autonomous offence. Previously, the French Penal Code traditionally prohibited incest as an aggravating circumstance based upon minority or the family relationship between the perpetrator and the victim. The problem was that, according to the strict construction principle, the judges had to determine the sexual offence, in turn supporting the aggravating circumstance of incest. To do so, they had to look for the evidence of the lack of consent determined by “violence, constraint, threaten or surprise”. The 2021 Act is the conclusion of several controversial reforms considering this issue.

Use of the best-interests test as the legal standard to justify medical treatment (or its cessation) in respect to legally incompetent adults or minors has come under sustained critique over the years. “Best interests” has variously been alleged to be indeterminate as well as susceptible to majoritarian ideology and inherent bias. It has also been alleged to be inferior to rights-based approaches. Against the background of several particularly hard cases involving minors discussed by Gillett in a prior article in this journal, this article considers some of these critiques. It concludes that these critical accounts make significant contributions to enabling a more procedurally and substantively robust consideration of what might be in a child’s best interests. However, it is argued that none of these accounts alone provide a superior framework that would justify jettisoning the concept of best interests. Further, it is suggested that best interests still has an important role in achieving patient-centered decisionmaking in this context. It concludes by suggesting a taxonomy of considerations when determining best interests.

Lower semi-continuity results for polyconvex functionals of the calculus of variations along sequences of maps u: Ω ⊂ ℝn → ℝm in W1,m, 2 ⩽ m⩽ n, weakly converging in W1,m-1, are established. In addition, for m = n + 1, we also consider the autonomous case for weakly converging maps in W1,n-1.

Professionals who are likely to come into contact with children play an essential role in the protection of children, thus we aimed to study the criteria they use to identify and report child sexual abuse cases. Based on the Factorial Survey design, we presented 974 Spanish (90%) and Latin American professionals from six fields (Psychology, Social Services, Education, Health, Law and Security) with hypothetical situations of sexual interaction with minors (systematically varying the type of sexual act, the child's and the other person's sex and age, the use of coercion and the type of strategy employed to involve the child), in order to examine their perception of abuse and willingness to report. According to results, the factors or criteria that most impact assessments are age asymmetry and use of coercion. Specifically, professionals are significantly more likely to perceive abuse and intend to report it if the other person involved in the interaction is much older than the minor and/or uses a coercive strategy, especially force, drugs or blackmail. Another relevant criterion is the type of sexual act, since acts involving intercourse, digital penetration or oral sex are significantly more likely to be deemed as abuse and reported.

We prove an upper bound on sums of squares of minors of $\{+1, -1\}$-matrices. The bound is sharp for Hadamard matrices, a result due to de Launey and Levin [‘$(1,-1)$-matrices with near-extremal properties’, SIAM J. Discrete Math.23(2009), 1422–1440], but our proof is simpler. We give several corollaries relevant to minors of Hadamard matrices.

The Iran-United States Claims Tribunal's recent decision in Sabet and The Islamic Republic of Iran sheds new light on difficult issues concerning the dual nationality of minors. In particular, the case was the first in which the Tribunal determined minor dual national claimants to have a dominant and effective nationality different from that of either of their parents. Further, the Tribunal broke new ground in its analysis of ‘the caveat,’ an equitable doctrine that can bar the claims of dual nationals. This article applauds the Tribunal's advances in its caveat jurisprudence and develops a new approach that would further those advances.