The number of removable edges in a 4-connected graph

AbstractLet G be a 4-connected graph. For an edge e of G, we do the following operations on G: first, delete the edge e from G, resulting the graph G−e; second, for all the vertices x of degree 3 in G−e, delete x from G−e and then completely connect the 3 neighbors of x by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by G⊖e. If G⊖e is still 4-connected, then e is called a removable edge of G. In this paper we prove that every 4-connected graph of order at least six (excluding the 2-cyclic graph of order six) has at least (4|G|+16)/7 removable edges. We also give the structural characterization of 4-connected graphs for which the lower bound is sharp

Divide and Adapt: Active Domain Adaptation via Customized Learning

Active domain adaptation (ADA) aims to improve the model adaptation performance by incorporating active learning (AL) techniques to label a maximally-informative subset of target samples. Conventional AL methods do not consider the existence of domain shift, and hence, fail to identify the truly valuable samples in the context of domain adaptation. To accommodate active learning and domain adaption, the two naturally different tasks, in a collaborative framework, we advocate that a customized learning strategy for the target data is the key to the success of ADA solutions. We present Divide-and-Adapt (DiaNA), a new ADA framework that partitions the target instances into four categories with stratified transferable properties. With a novel data subdivision protocol based on uncertainty and domainness, DiaNA can accurately recognize the most gainful samples. While sending the informative instances for annotation, DiaNA employs tailored learning strategies for the remaining categories. Furthermore, we propose an informativeness score that unifies the data partitioning criteria. This enables the use of a Gaussian mixture model (GMM) to automatically sample unlabeled data into the proposed four categories. Thanks to the "divideand-adapt" spirit, DiaNA can handle data with large variations of domain gap. In addition, we show that DiaNA can generalize to different domain adaptation settings, such as unsupervised domain adaptation (UDA), semi-supervised domain adaptation (SSDA), source-free domain adaptation (SFDA), etc.Comment: CVPR2023, Highlight pape