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We verify a recent conjecture of Sehrawat, Kumar and Ahlawat on the minimum bisection width (a notion that was rediscovered in 2020 by Acharya and Kureethara under the name of rna number) of powers of cycles., Comment: 3 pages

The $d$-Cut problem is to decide if a graph has an edge cut such that each vertex has at most $d$ neighbours at the opposite side of the cut. If $d=1$, we obtain the intensively studied Matching Cut problem. The $d$-Cut problem has been studied as well, but a systematic study for special graph classes was lacking. We initiate such a study and consider classes of bounded diameter, bounded radius and $H$-free graphs. We prove that for all $d\geq 2$, $d$-Cut is polynomial-time solvable for graphs of diameter $2$, $(P_3+P_4)$-free graphs and $P_5$-free graphs. These results extend known results for $d=1$. However, we also prove several NP-hardness results for $d$-Cut that contrast known polynomial-time results for $d=1$. Our results lead to full dichotomies for bounded diameter and bounded radius and to almost-complete dichotomies for $H$-free graphs.

In recent years, convolutional neural networks for semantic segmentation of breast ultrasound (BUS) images have shown great success; however, two major challenges still exist. 1) Most current approaches inherently lack the ability to utilize tissue anatomy, resulting in misclassified image regions. 2) They struggle to produce accurate boundaries due to the repeated down-sampling operations. To address these issues, we propose a novel breast anatomy-aware network for capturing fine image details and a new smoothness term that encodes breast anatomy. It incorporates context information across multiple spatial scales to generate more accurate semantic boundaries. Extensive experiments are conducted to compare the proposed method and eight state-of-the-art approaches using a BUS dataset with 325 images. The results demonstrate the proposed method significantly improves the segmentation of the muscle, mammary, and tumor classes and produces more accurate fine details of tissue boundaries.

A matching cut in a graph G is an edge cut of G that is also a matching. This short survey gives an overview of old and new results and open problems for Maximum Matching Cut, which is to determine the size of a largest matching cut in a graph. We also compare this problem with the related problems Matching Cut, Minimum Matching Cut, and Perfect Matching Cut, which are to determine if a graph has a matching cut; the size of a smallest matching cut in a graph; and if a graph has a matching cut that is a perfect matching, respectively. Moreover, we discuss a relationship between Maximum Matching Cut and Max Cut, which is to determine the size of a largest edge cut in a graph, as well as a relationship between Minimum Matching Cut and Min Cut, which is to determine the size of a smallest edge cut in a graph.

In this paper, we consider the following two problems: (i) Deletion Blocker($\alpha$) where we are given an undirected graph $G=(V,E)$ and two integers $k,d\geq 1$ and ask whether there exists a subset of vertices $S\subseteq V$ with $|S|\leq k$ such that $\alpha(G-S) \leq \alpha(G)-d$, that is the independence number of $G$ decreases by at least $d$ after having removed the vertices from $S$; (ii) Transversal($\alpha$) where we are given an undirected graph $G=(V,E)$ and two integers $k,d\geq 1$ and ask whether there exists a subset of vertices $S\subseteq V$ with $|S|\leq k$ such that for every maximum independent set $I$ we have $|I\cap S| \geq d$. We show that both problems are polynomial-time solvable in the class of co-comparability graphs by reducing them to the well-known Vertex Cut problem. Our results generalize a result of [Chang et al., Maximum clique transversals, Lecture Notes in Computer Science 2204, pp. 32-43, WG 2001] and a recent result of [Hoang et al., Assistance and interdiction problems on interval graphs, Discrete Applied Mathematics 340, pp. 153-170, 2023].

In healthcare, the ability to care for oneself is reflected in the "Activities of Daily Living (ADL)," which serve as a measure of functional ability (functioning). A lack of functioning may lead to poor living conditions requiring personal care and assistance. To accurately identify those in need of support, assistance programs continuously evaluate participants' functioning across various domains. However, the assessment process may encounter consistency issues when multiple assessors with varying levels of expertise are involved. Novice assessors, in particular, may lack the necessary preparation for real-world interactions with participants. To address this issue, we developed a dialogue system that simulates interactions between assessors and individuals of varying functioning in a natural and reproducible way. The dialogue system consists of two major modules, one for natural language understanding (NLU) and one for natural language generation (NLG), respectively. In order to generate responses consistent with the underlying knowledge base, the dialogue system requires both an understanding of the user's query and of biographical details of an individual being simulated. To fulfill this requirement, we experimented with query classification and generated responses based on those biographical details using some recently released InstructGPT-like models., Comment: Accepted to ACL 2023 DialDoc Workshop

The (Perfect) Matching Cut problem is to decide if a graph $G$ has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of $G$. Both Matching Cut and Perfect Matching Cut are known to be NP-complete. A perfect matching cut is also a matching cut with maximum number of edges. To increase our understanding of the relationship between the two problems, we perform a complexity study for the Maximum Matching Cut problem, which is to determine a largest matching cut in a graph. Our results yield full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and $H$-free graphs. A disconnected perfect matching of a graph $G$ is a perfect matching that contains a matching cut of $G$. We also show how our new techniques can be used for finding a disconnected perfect matching with a largest matching cut for special graph classes. In this way we can prove that the decision problem Disconnected Perfect Matching is polynomial-time solvable for $(P_6+sP_2)$-free graphs for every $s\geq 0$, extending a known result for $P_5$-free graphs (Bouquet and Picouleau, 2020).