arXiv subjects

Jianbing Liu

Publications and source records attributed to Jianbing Liu.

8 recordsLinked to original sources

arXiv2026

Partial Petrial Polynomials of Bouquets

Gross, Mansour, and Tucker [European J. Combin., 95 (2021): 103329] introduced the \emph{partial Petrial polynomial} of a ribbon graph $G$, denoted by $^{\partial}{\varepsilon^{\times}_G}(z)$. For a prime bouquet $B_n$, Yan and Li [Discrete Appl. Math., 375 (2025): 281-289] determined $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ when the intersection graph $I(B_n)$ is either the complete graph or a path, and provided an equivalent condition under which the lowest degree of the nonzero coefficient in $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ is $1$. In this paper, we determine $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ when the intersection graph is a cycle. Moreover, we present a complete characterization of the prime bouquets whose lowest nonzero term in $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ is of degree $2$ and determine the partial Petrial polynomial for the prime bouquets. As corollaries, we determine $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ when $I(B_n)$ is the complete bipartite and tripartite graph.

math.CO↗

arXiv2026

Partial Petrial Polynomials of Ribbon Graphs

Gross, Mansour, and Tucker [European J. Combin., 95 (2021): 103329] introduced the \emph{partial Petrial polynomial} of a ribbon graph $G$, denoted by $^{\partial}{\varepsilon^{\times}_{G}}(z)$. Beck and Mellor proved, in both orientable and non-orientable cases respectively, that the Euler genus of a bouquet equals the rank of a certain matrix over $\mathbb{GF}(2)$. In this paper, we first generalize Beck and Mellor's results from bouquets to all ribbon graphs. Secondly, we give an equivalent representation of the partial Petrial polynomial for all ribbon graphs. Specifically, the partial Petrial polynomial of a ribbon graph $G$ with $n$ vertices is equal to the sum of this polynomial for $2^{n-1}$ distinct bouquets. Moreover, we give the definition of a modified partial Petrial polynomial by assigning coefficients $+1$ or $-1$ to the terms in the partial Petrial polynomial such that the resulting polynomial satisfies the four-term relation for graphs. Finally, we generalize the modified partial Petrial polynomial from bouquets to all signed simple graphs and prove that this polynomial is $4$-invariant, which provides an answer to the problem posed by Lando [J.~Combin.~Theory Ser.~B,~80~(1) (2000): 104-121]: Which of the known graph invariants are $4$-invariants?

math.CO↗

arXiv2026

A DMD-Based Adaptive Modulation Method for High Dynamic Range Imaging in High-Glare Environments

Background The accuracy of photomechanics measurements critically relies on image quality,particularly under extreme illumination conditions such as welding arc monitoring and polished metallic surface analysis. High dynamic range (HDR) imaging above 120 dB is essential in these contexts. Conventional CCD/CMOS sensors, with dynamic ranges typically below 70 dB, are highly susceptible to saturation under glare, resulting in irreversible loss of detail and significant errors in digital image correlation (DIC). Methods This paper presents an HDR imaging system that leverages the spatial modulation capability of a digital micromirror device (DMD). The system architecture enables autonomous regional segmentation and adaptive exposure control for high-dynamic-range scenes through an integrated framework comprising two synergistic subsystems: a DMD-based optical modulation unit and an adaptive computational imaging pipeline. Results The system achieves a measurable dynamic range of 127 dB, effectively eliminating satu ration artifacts under high glare. Experimental results demonstrate a 78% reduction in strain error and improved DIC positioning accuracy, confirming reliable performance across extreme intensity variations. Conclusion The DMD-based system provides high fidelity adaptive HDR imaging, overcoming key limitations of conventional sensors. It exhibits strong potential for optical metrology and stress analysis in high-glare environments where traditional methods are inadequate.

cs.CV↗

arXiv2025

Oriented Diameter of Mixed Graphs with Given Maximum Undirected Degree

In 2018, Dankelmann, Gao, and Surmacs [J. Graph Theory, 88(1): 5--17, 2018] established sharp bounds on the oriented diameter of a bridgeless undirected graph and a bridgeless undirected bipartite graph in terms of vertex degree. In this paper, we extend these results to \emph{mixed graphs}, which contain both directed and undirected edges. Let the \emph{undirected degree} $d^*_G(x)$ of a vertex $x \in V(G)$ be the number of its incident undirected edges in a mixed graph $G$ of order $n$, and let the \emph{maximum undirected degree} be $Δ^*(G) = \max\{d^*_G(v) : v \in V(G)\}$. We prove that \begin{align*} \text{(1)}\quad & \overrightarrow{\mathrm{diam}}(G) \leq n - Δ^* + 3 && \text{if $G$ is undirected, or contains a vertex $u$ with $d^*_G(u) = Δ^*$} \\ & && \text{and $d^+_G(u) + d^-_G(u) \geq 2$, or $Δ^* = 5$ and $d^+_G(u) + d^-_G(u) = 1$;} \\ \text{(2)}\quad & \overrightarrow{\mathrm{diam}}(G) \leq n - Δ^* + 4 && \text{otherwise}. \end{align*} We also establish bounds for mixed bipartite graphs. If $G$ is a bridgeless mixed bipartite graph with partite sets $A$ and $B$, and $u \in B$, then \begin{align*} \text{(1)}\quad & \overrightarrow{\mathrm{diam}}(G) \leq 2(|A| - d(u)) + 7 && \text{if $G$ is undirected;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \text{(2)}\quad & \overrightarrow{\mathrm{diam}}(G) \leq 2(|A| - d^*(u)) + 8 && \text{if $d^+_G(u) + d^-_G(u) \geq 2$;} \\ \text{(3)}\quad & \overrightarrow{\mathrm{diam}}(G) \leq 2(|A| - d^*(u)) + 10 && \text{otherwise}. \end{align*} All of the above bounds are sharp, except possibly the last one.

math.CO↗

arXiv2025

Event-based multi-view photogrammetry for high-dynamic, high-velocity target measurement

The characterization of mechanical properties for high-dynamic, high-velocity target motion is essential in industries. It provides crucial data for validating weapon systems and precision manufacturing processes etc. However, existing measurement methods face challenges such as limited dynamic range, discontinuous observations, and high costs. This paper presents a new approach leveraging an event-based multi-view photogrammetric system, which aims to address the aforementioned challenges. First, the monotonicity in the spatiotemporal distribution of events is leveraged to extract the target's leading-edge features, eliminating the tailing effect that complicates motion measurements. Then, reprojection error is used to associate events with the target's trajectory, providing more data than traditional intersection methods. Finally, a target velocity decay model is employed to fit the data, enabling accurate motion measurements via ours multi-view data joint computation. In a light gas gun fragment test, the proposed method showed a measurement deviation of 4.47% compared to the electromagnetic speedometer.

cs.CV↗

arXiv2025

Complexity and Algorithm for the Matching vertex-cutset Problem

In 1985, Chvátal introduced the concept of star cutsets as a means to investigate the properties of perfect graphs, which inspired many researchers to study cutsets with some specific structures, for example, star cutsets, clique cutsets, stable cutsets. In recent years, approximation algorithms have developed rapidly, the computational complexity associated with determining the minimum vertex cut possessing a particular structural property have attracted considerable academic attention. In this paper, we demonstrate that determining whether there is a matching vertex-cutset in $H$ with size at most $k$, is $\mathbf{NP}$-complete, where $k$ is a given positive integer and $H$ is a connected graph. Furthermore, we demonstrate that for a connected graph $H$, there exists a $2$-approximation algorithm in $O(nm^2)$ for us to find a minimum matching vertex-cutset. Finally, we show that every plane graph $H$ satisfying $H\not\in\{K_2, K_4\}$ contains a matching vertex-cutset with size at most three, and this bound is tight.

cs.DS↗

arXiv2024

Diameter two orientability of mixed graphs

In 1967, Katona and Szemerédi showed that no undirected graph with $n$ vertices and fewer than $\frac{n}{2}\log_2\frac{n}{2}$ edges admits an orientation of diameter two. In 1978, Chvátal and Thomassen revealed the complexity of determining whether an undirected graph can be oriented to achieve a diameter of two, proving it to be NP-complete. This breakthrough has sparked ongoing interest in identifying sufficient conditions for graphs to be oriented with the smallest possible diameter of two -- critical for optimizing communication and network flow in larger structures. In 2019, Czabarka, Dankelmann, and Székely significantly advanced this field by establishing that the minimum degree threshold for achieving such an orientation in undirected graphs of order $n$ is $\frac{n}{2} + Θ(\ln n)$. In this paper, we extend this foundational result by determining the minimum degree threshold necessary for realizing an orientation with diameter two in mixed graphs, which contain both undirected and directed edges. Mixed graphs offer a versatile framework, representing an intermediate stage in the orientation process, making our findings a substantial generalization of previous results.

math.CO↗

arXiv2024

A Note on Improved bounds for the Oriented Radius of Mixed Multigraphs

For a positive integer $r$, let $f(r)$ denote the smallest number such that any 2-edge connected mixed graph with radius $r$ has an oriented radius of at most $f(r)$. Recently, Babu, Benson, and Rajendraprasad significantly improved the upper bound of $f(r)$ by establishing that $f(r) \leq 1.5r^2 + r + 1$, see [Improved bounds for the oriented radius of mixed multigraphs, J. Graph Theory, 103 (2023), 674-689]. Additionally, they demonstrated that if each edge of a graph $G$ is contained within a cycle of length at most $η$, then the oriented radius of $G$ is at most $1.5rη$. The authors' results were derived through Observation 1, which served as the foundation for the development of Algorithm ORIENTOUT and Algorithm ORIENTIN. By integrating these algorithms, they obtained the improved bounds. However, an error has been identified in Observation 1, necessitating revisions to Algorithm ORIENTOUT and Algorithm ORIENTIN. In this note, we address the error and propose the necessary modifications to both algorithms, thereby ensuring the correctness of the conclusions.

math.CO↗