Dieses Lehrbuch vermittelt die Grundlagen und Konzepte der modernen Kombinatorik in anschaulicher Weise. Die verständliche Darlegung richtet sich an Studierende der Mathematik, der Naturwissenschaften

A graph G is H-free if any subset of V(G) does not induce a subgraph of G that is isomorphic to H. Given a graph H, we present sufficient and necessary conditions for a graph G such that G/e is H-free for any edge e in E(G). Thereafter, we use these conditions to characterize forests, claw-free, 2K_2-free, C_4-free, C_5-free, split, and pseudo-split graphs.

Given a family of graphs ℋ, a graph G is ℋ-free if any subset of V(G) does not induce a subgraph of G that is isomorphic to any graph in ℋ. We present sufficient and necessary conditions for a graph G such that G/e is ℋ-free for any edge e in E(G). Thereafter, we use these conditions to characterize claw-free and line graphs.

The neighborhood polynomial of graph G is the generating function for the number of vertex subsets of G of which the vertices have a common neighbor in G. In this paper, we investigate the behavior of this polynomial under several graph operations. Specifically, we provide an explicit formula for the neighborhood polynomial of the graph obtained from a given graph G by vertex attachment. We use this result to propose a recursive algorithm for the calculation of the neighborhood polynomial. Finally, we prove that the neighborhood polynomial can be found in polynomial-time in the class of k-degenerate graphs.

No AccessGraphentheorieDec 2021Planare Graphen - die Eulersche PolyederformelPeter TittmannPeter Tittmannhttps://doi.org/10.3139/9783446472471.003SectionsAboutPDF ToolsAdd to FavoritesDownload CitationTrack CitationsCopy LTI LinkPDF key 'share (en)' returned an object instead of string.FacebookTwitterEmailLinkedIn previous chapternext chapter FiguresReferencesRelatedDetails 2021Pages: 44-59Print ISBN: 978-3-446-47196-2eISBN: 978-3-446-47247-1 Copyright & Permissions© 2022 Carl Hanser Verlag GmbH & Co. KGPDF downloadLoading ...

Crapo introduced a construction of interval partitions of the Boolean lattice for sets equipped with matroid structure. This construction, in the context of graphic matroids, is related to the notion of edge activities introduced by Tutte. This implies that each spanning subgraph of a connected graph can be constructed from edges of exactly one spanning tree by deleting a unique subset of internally active edges and adding a unique subset of externally active edges. Since the family of vertex independent sets does not give rise to a matroid structure we can not apply Crapo's construction on the vertex set when using the family of independent sets as generating sets. In this paper, we introduce the concept of vertex activities to tackle the problem of generating interval partitions of the Boolean lattice of the vertex set. We show how to generate a cover, present some properties related to vertex activities of some special maximal independent sets and consider some special graphs. Finally, we will show that level labellings in pruned graphs always generate a partition.

Regional tree die-off events generate large quantities of standing dead wood, raising concern over catastrophic wildfire and other hazards. Governmental responses to tree die-off have often focused on incentivizing biomass energy production that utilizes standing dead trees removed for safety concerns. However, the full distribution of potential woody bioenergy feedstock after tree die-off has not been evaluated due to the complexities of surveying and precisely measuring large forested areas. In this paper, we present a novel method for estimating standing dead biomass at a fine spatial resolution that combines aerial survey data with forest structure maps. Using this method, we quantify biomass generated by the unprecedented tree die-off that occurred in California following a 4-year drought and widespread pest outbreaks. The results are used to estimate feasibly recoverable feedstock for energy production. We find that approximately 95.1 million bone-dry tons (BDT) of dead biomass resulted from 2012–2017 mortality, with a lower bound of 26.2 million BDT. In other words, of the aboveground live tree biomass in 2012, ~1.3–4.8% died by 2017. Of the standing dead biomass, 29% meets minimum constraints for potential cost-effective bioenergy feedstock. This proportion drops to as low as 15% in the most affected areas due to terrain slope, wilderness status, and other factors, highlighting the need to complement disposal via biomass energy with other strategies to mitigate the risks of the tree mortality crisis, which is likely to only become more severe over time due to climate change.

No AccessGraphentheorieDec 2021Der Zusammenhang von GraphenPeter TittmannPeter Tittmannhttps://doi.org/10.3139/9783446472471.006SectionsAboutPDF ToolsAdd to FavoritesDownload CitationTrack CitationsCopy LTI LinkPDF key 'share (en)' returned an object instead of string.FacebookTwitterEmailLinkedIn previous chapternext chapter FiguresReferencesRelatedDetails 2021Pages: 94-108Print ISBN: 978-3-446-47196-2eISBN: 978-3-446-47247-1 Copyright & Permissions© 2022 Carl Hanser Verlag GmbH & Co. KGPDF downloadLoading ...

The neighborhood polynomial of graphG is the generating function for the number of vertex subsets of G of which the vertices have a common neighbor in G. In this paper, we investigate the behavior of this polynomial under several graph operations. Specifically, we provide an explicit formula for the neighborhood polynomial of the graph obtained from a given graph G by vertex attachment. We use this result to propose a recursive algorithm for the calculation of the neighborhood polynomial. Finally, we prove that the neighborhood polynomial can be found in polynomial-time in the class of k-degenerate graphs.

The neighborhood complex of a graph is the family of subsets of open neighborhoods of its vertices. The neighborhood polynomial is the ordinary generating function for the number of sets of the neighborhood complex with respect to their cardinality. This paper provides a new representation of the neighborhood polynomial as a sum over complete bipartite subgraphs of a graph. Using the close relation between the domination polynomial and the neighborhood polynomial of a graph, we can also give a new presentation of the domination polynomial. Finally we show that finding the number of certain double cliques of a graph is sufficient to determine the number of dominating sets of a graph.

The reliability polynomial of a finite undirected graph gives the probability that the operational edges of induce a connected graph assuming that all edges of fail independently with identical probability . In this article we investigate the probability that the operational edges of a graph with randomly failing edges induce a biconnected or two‐edge connected subgraph, which corresponds to demands for redundancy or higher throughput in communication networks. The computation of the biconnected or two‐edge connected reliability for general graphs is computationally intractable (#P‐hard). We provide recurrence relations for biconnected and two‐edge connected reliability of complete graphs. As a consequence, we can determine the number of biconnected and two‐edge connected graphs with given order and size. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 69(4), 408–414 2017

Network reliability analysis has interesting applications in areas such as computer and mobile networks. However, the computation of many important reliability measures (all-terminal reliability, reachability) turns out to be NP-hard. This statement applies to the computation of relevant reliability importance measures, too. In this paper we introduce local importance measures that describe the importance of an edge or vertex of the network in its local network neighborhood. Suitable scaling of the local neighborhood renders the computation of generally intractable reliability measures possible.

Counting dominating sets in a graph $G$ is closely related to the neighborhood complex of $G$. We exploit this relation to prove that the number of dominating sets $d(G)$ of a graph is determined by the number of complete bipartite subgraphs of its complement. More precisely, we state the following. Let $G$ be a simple graph of order $n$ such that its complement has exactly $a(G)$ subgraphs isomorphic to $K_{2p,2q}$ and exactly $b(G)$ subgraphs isomorphic to $K_{2p+1,2q+1}$. Then $d(G) = 2^n -1 + 2[a(G)-b(G)]$. We also show some new relations between the domination polynomial and the neighborhood polynomial of a graph.

The bipartition polynomial of a graph is a generalization of many other graph polynomials, including the domination, Ising, matching, independence, cut, and Euler polynomial. We show in this paper that it is also a powerful tool for proving graph properties. In addition, we can show that the bipartition polynomial is polynomially reconstructible, which means that we can recover it from the multiset of bipartition polynomials of one-edge-deleted subgraphs.