A novel extension of the classical signal -adaptive median lter (SAM) is proposed in this paper. It is the so-called morphological signal -adaptive median lter (MSAM). Two mod-iications are introduced in the SAM lter aiming at: (1) enhancing SAM impulse detection mechanism so that it detects not only impulses of a constant amplitude but randomly-valued impulses as well, (2) employing an anisotr...

Tree-like partial cubes were introduced in [B. Brešar, W. Imrich, S. Klavžar, Tree-like isometric subgraphs of hypercubes, Discuss. Math. Graph Theory, 23 (2003), 227–240] as a generalization of median graphs. We present some incorrectnesses from that article. In particular we point to a gap in the proof of the theorem about the dismantlability of the cube graph of a tree-like partial cube and ...

Geodetic sets in graphs are briefly surveyed. After an overview of earlier results, we concentrate on recent studies of the geodetic number and related invariants in graphs. Geodetic sets in Cartesian products of graphs and in median graphs are considered in more detail. Algorithmic issues and relations with several other concepts, arising from various convex and interval structures in graphs, ...

Convex cycles play a role e.g. in the context of product graphs. We introduce convex cycle bases and describe a polynomial-time algorithm that recognizes whether a given graph has a convex cycle basis and provides an explicit construction in the positive case. Relations between convex cycles bases and other types of cycles bases are discussed. In particular we show that if G has a unique minima...

Aprofileπ = (x1, . . . , xk), of length k, in a finite connected graphG is a sequence of vertices ofG, with repetitions allowed. Amedian x ofπ is a vertex forwhich the sumof the distances from x to the vertices in the profile is minimum. The median function finds the set of all medians of a profile. Medians are important in location theory and consensus theory. A median graph is a graph for whi...

Given a set Σ of bipartitions of some finite set X of cardinality at least 2, one can associate to Σ a canonical X-labeled graph B(Σ), called the Buneman graph. This graph has several interesting mathematical properties — for example, it is a median network and therefore an isometric subgraph of a hypercube. It is commonly used as a tool in studies of DNA sequences gathered from populations. In...

Isometric subgraphs of hypercubes, or partial cubes as they are also called, are a rich class of graphs that include median graphs, subdivision graphs of complete graphs, and classes of graphs arising in mathematical chemistry and biology. In general one can recognize whether a graph on n vertices and m edges is a partial cube in O(mn) steps, faster recognition algorithms are only known for med...

This paper considers the problem of finding a meaningful template function that represents the common pattern of a sample of curves. To address this issue, a novel algorithm based on a robust version of the isometric featuring mapping (Isomap) algorithm is developed. Assuming that the functional data lie on an intrinsically low-dimensional smooth manifold with unknown underlying structure, we p...

We address the problem of finding a 1-median on a cactus graph. The problem has already been solved in linear time by the algorithms of Burkard and Krarup (1998), and Lan and Wang (2000). These algorithms are complicated and need efforts. Hence, we develop in this paper a simpler algorithm. First, we construct a condition for a cycle that contains a 1-median or for a vertex that is indeed a 1-m...

For stable marriage (SM) and solvable stable roommates (SR) instances, it is known that there are stable matchings that assign each participant to his or her (lower/upper) median stable partner. Moreover, for SM instances, a stable matching has this property if and only if it is a median of the distributive lattice formed by the instance’s stable matchings. In this paper, we show that the above...