707 research outputs found

    Characterizing Block Graphs in Terms of their Vertex-Induced Partitions

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    Given a finite connected simple graph G=(V,E)G=(V,E) with vertex set VV and edge set EβŠ†(V2)E\subseteq \binom{V}{2}, we will show that 1.1. the (necessarily unique) smallest block graph with vertex set VV whose edge set contains EE is uniquely determined by the VV-indexed family PG:=(Ο€0(G(v)))v∈V{\bf P}_G:=\big(\pi_0(G^{(v)})\big)_{v \in V} of the various partitions Ο€0(G(v))\pi_0(G^{(v)}) of the set VV into the set of connected components of the graph G(v):=(V,{e∈E:vβˆ‰e})G^{(v)}:=(V,\{e\in E: v\notin e\}), 2.2. the edge set of this block graph coincides with set of all 22-subsets {u,v}\{u,v\} of VV for which uu and vv are, for all w∈Vβˆ’{u,v}w\in V-\{u,v\}, contained in the same connected component of G(w)G^{(w)}, 3.3. and an arbitrary VV-indexed family Pp=(pv)v∈V{\bf P}p=({\bf p}_v)_{v \in V} of partitions Ο€v\pi_v of the set VV is of the form Pp=PpG{\bf P}p={\bf P}p_G for some connected simple graph G=(V,E)G=(V,E) with vertex set VV as above if and only if, for any two distinct elements u,v∈Vu,v\in V, the union of the set in pv{\bf p}_v that contains uu and the set in pu{\bf p}_u that contains vv coincides with the set VV, and {v}∈pv\{v\}\in {\bf p}_v holds for all v∈Vv \in V. As well as being of inherent interest to the theory of block graphs, these facts are also useful in the analysis of compatible decompositions and block realizations of finite metric spaces

    Characterizing block graphs in terms of their vertex-induced partitions

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    Block graphs are a generalization of trees that arise in areas such as metric graph theory, molecular graphs, and phylogenetics. Given a finite connected simple graph G=(V,E)G=(V,E) with vertex set VV and edge set EβŠ†(V2)E\subseteq \binom{V}{2}, we will show that the (necessarily unique) smallest block graph with vertex set VV whose edge set contains EE is uniquely determined by the VV-indexed family \Pp_G =\big(\pi_v)_{v \in V} of the partitions Ο€v\pi_v of the set VV into the set of connected components of the graph (V,{e∈E:vβˆ‰e})(V,\{e\in E: v\notin e\}). Moreover, we show that an arbitrary VV-indexed family \Pp=(\p_v)_{v \in V} of partitions \p_v of the set VV is of the form \Pp=\Pp_G for some connected simple graph G=(V,E)G=(V,E) with vertex set VV as above if and only if, for any two distinct elements u,v∈Vu,v\in V, the union of the set in \p_v that contains uu and the set in \p_u that contains vv coincides with the set VV, and \{v\}\in \p_v holds for all v∈Vv \in V. As well as being of inherent interest to the theory of block graphs,these facts are also useful in the analysis of compatible decompositions of finite metric spaces

    Landscapes of data sets and functoriality of persistent homology

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    The aim of this article is to describe a new perspective on functoriality of persistent homology and explain its intrinsic symmetry that is often overlooked. A data set for us is a finite collection of functions, called measurements, with a finite domain. Such a data set might contain internal symmetries which are effectively captured by the action of a set of the domain endomorphisms. Different choices of the set of endomorphisms encode different symmetries of the data set. We describe various category structures on such enriched data sets and prove some of their properties such as decompositions and morphism formations. We also describe a data structure, based on coloured directed graphs, which is convenient to encode the mentioned enrichment. We show that persistent homology preserves only some aspects of these landscapes of enriched data sets however not all. In other words persistent homology is not a functor on the entire category of enriched data sets. Nevertheless we show that persistent homology is functorial locally. We use the concept of equivariant operators to capture some of the information missed by persistent homology

    Fault tolerance in space-based digital signal processing and switching systems: Protecting up-link processing resources, demultiplexer, demodulator, and decoder

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    Fault tolerance features in the first three major subsystems appearing in the next generation of communications satellites are described. These satellites will contain extensive but efficient high-speed processing and switching capabilities to support the low signal strengths associated with very small aperture terminals. The terminals' numerous data channels are combined through frequency division multiplexing (FDM) on the up-links and are protected individually by forward error-correcting (FEC) binary convolutional codes. The front-end processing resources, demultiplexer, demodulators, and FEC decoders extract all data channels which are then switched individually, multiplexed, and remodulated before retransmission to earth terminals through narrow beam spot antennas. Algorithm based fault tolerance (ABFT) techniques, which relate real number parity values with data flows and operations, are used to protect the data processing operations. The additional checking features utilize resources that can be substituted for normal processing elements when resource reconfiguration is required to replace a failed unit
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