No abstract

The dissociation number diss ( G ) $\text{diss}(G)$ of a graph G $G$ is the maximum order of a set of vertices of G $G$ inducing a subgraph that is of maximum degree at most 1. Computing the dissociation number of a given graph is algorithmically hard even when restricted to subcubic bipartite graphs. For a graph G $G$ with n $n$ vertices, m $m$ edges, k $k$ components, and c 1 ${c}_{1}$ induced cycles of length 1 modulo 3, we show diss ( G ) ≥ n − 1 3 ( m + k + c 1 ) $\text{diss}(G)\ge n-\frac{1}{3}(m+k+{c}_{1})$. Furthermore, we characterize the extremal graphs in which every two cycles are vertex‐disjoint.

No abstract

X-ray fluorescence spectrometry (XRF) is a technique that allows determining non-destructively the composition of elements within a sample. Focussing the excitation X-ray beam to a small spot that is moved...

The independence number and the dissociation number of a graph are the largest orders of induced subgraphs of of maximum degree at most 0 and at most 1, respectively. We consider possible improvements of the obvious inequality . For connected cubic graphs distinct from , we show , and describe the rich and interesting structure of the extremal graphs in detail. For bipartite graphs, and, more generally, triangle‐free graphs, we also obtain improvements. For subcubic graphs though, the inequality cannot be improved in general, and we characterize all extremal subcubic graphs.