Linear-Time Recognition of Double-Threshold Graphs

A graph $$G = (V,E)$$ G = ( V , E ) is a double-threshold graph if there exist a vertex-weight function $$w :V \rightarrow \mathbb {R}$$ w : V → R and two real numbers $$\mathtt {lb}, \mathtt {ub}\in \mathbb {R}$$ lb , ub ∈ R such that $$uv \in E$$ u v ∈ E if and only if $$\mathtt {lb}\le \mathtt {w}(u) + \mathtt {w}(v) \le \mathtt {ub}$$ lb ≤ w ( u ) + w ( v ) ≤ ub . In the literature, those graphs are studied also as the pairwise compatibility graphs that have stars as their underlying trees. We give a new characterization of double-threshold graphs that relates them to bipartite permutation graphs. Using the new characterization, we present a linear-time algorithm for recognizing double-threshold graphs. Prior to our work, the fastest known algorithm by Xiao and Nagamochi [Algorithmica 2020] ran in $$O(n^{3} m)$$ O ( n 3 m ) time, where n and m are the numbers of vertices and edges, respectively.

Water Extraction in SAR Images Using Features Analysis and Dual-Threshold Graph Cut Model

Timely identifying and detecting water bodies from SAR images are significant for flood monitoring and water resources management. In recent decades, deep learning has been applied to water extraction but is subject to the large difficulty of acquiring SAR dataset of various water bodies types, as well as heavy labeling work. In addition, the traditional methods mostly occur over the large, open lakes and rivers, rarely focusing on complex areas such as the urban water, and cannot automatically acquire the classification threshold. To address these issues, a novel water extraction method is proposed with high accuracy in this paper. Firstly, a multiscale feature extraction using a Gabor filter is conducted to reduce the noise and roughly identify water feature. Secondly, we apply the Otsu algorithm as well as a voting strategy to initially extract the homogeneous regions and for subsequent Gaussian mixture model (GMM). Finally, the dual threshold is obtained from the fitted Gaussian distribution of water and non-water, which is integrated into the graph cut model to redefine the weights of the edges, then constructing the energy function of the water map. The dual-threshold graph cut (DTGC) model precisely pinpoints the water location by minimizing the energy function. To verify the efficiency and robustness, our method and comparison methods, including the IGC method and IACM method, are tested on six different types of water bodies, by performing the accuracy assessment via comparing outcomes with the manually labeled ground truth. The qualitative and quantitative results show that the overall accuracy of our method for the whole dataset all surpasses 99%, along with an obvious improvement of the Kappa, F1-score, and IoU indicators. Therefore, DTGC method has the absolute advantage of automatically capturing water maps in different scenes of SAR images without specific prior knowledge and can also determine the optimal threshold range.

Graph Coloring of Symptom Networks

The study of graph coloring is especially timely given that some of the most popular models and methods for estimating symptom networks have theoretical ties to graph coloring. We use graph-coloring algorithms to generate all possible colorings of the complement of a threshold graph associated with a symptom network. This facilitates the identification of symptoms that have flexibility in their color assignments, which often indicates that they serve as a bridge between multiple cohesive subsets of symptoms. Two published symptom networks are used for demonstration purposes.

Homomorphisms into Loop-Threshold Graphs

Many problems in extremal graph theory correspond to questions involving homomorphisms into a fixed image graph. Recently, there has been interest in maximizing the number of homomorphisms from graphs with a fixed number of vertices and edges into small image graphs. For the image graph $H_\text{ind}$, the graph on two adjacent vertices, one of which is looped, each homomorphism from $G$ to $H_\text{ind}$ corresponds to an independent set in $G$. It follows from the Kruskal-Katona theorem that the number of homomorphisms to $H_\text{ind}$ is maximized by the lex graph, whose edges form an initial segment of the lex order. A loop-threshold graph is a graph built recursively from a single vertex, which may be looped or unlooped, by successively adding either a looped dominating vertex or an unlooped isolated vertex at each stage. Thus, the graph $H_\text{ind}$ is a loop-threshold graph. We survey known results for maximizing the number of homomorphisms into small loop-threshold image graphs. The only extremal homomorphism problem with a loop-threshold image graph on at most three vertices not yet solved is $H_\text{ind}\cup E_1$, where extremal graphs are the union of a lex graph and an empty graph. The only question that remains is the size of the lex component of the extremal graph. While we cannot give an exact answer for every number of vertices and edges, we establish the significance of and give bounds on $\ell(m)$, the number of vertices in the lex component of the extremal graph with $m$ edges and at least $m+1$ vertices.

A note on the eigenvalue free intervals of some classes of signed threshold graphs

We consider a particular class of signed threshold graphs and their eigenvalues. If Ġ is such a threshold graph and Q(Ġ ) is a quotient matrix that arises from the equitable partition of Ġ , then we use a sequence of elementary matrix operations to prove that the matrix Q(Ġ ) – xI (x ∈ ℝ) is row equivalent to a tridiagonal matrix whose determinant is, under certain conditions, of the constant sign. In this way we determine certain intervals in which Ġ has no eigenvalues.