graph-theory

The Quest Begins (The “Why”) I still remember the first time I stared at a whiteboard interview question that asked for the shortest number of moves a knight needs to reach a target square on a chessboard. My brain went into panic mode: “Do I try every possible path? Do I keep a visited set? Do I… recurse?” I felt like Neo in The Matrix before he sees the code—everything was a blur of green symbo…

Q1) 547. Number of Provinces 👉 LeetCode Problem Link 💡 Initial Idea & The Mistake My first instinct was to simply count the rows where the array sum equals 1 (meaning a city is only connected to itself), assuming everything else would naturally hook together into a single province. The flaw in that logic: I completely overlooked scenarios where you might have multiple distinct, disconnected clust…

Let $ U(\mathbb{R}^2) $ be the unit distance graph of the plane (vertices = points in $\mathbb{R}^2$, edges = pairs at Euclidean distance exactly 1). The chromatic number $\chi(U)$ is known to be ...

Each of several possible definitions of local injectivity for a homomorphism of an oriented graph $G$ to an oriented graph $H$ leads to an injective oriented colouring problem. For each case in which such a problem is solvable in polynomial time, we identify a set $\mathcal{F}$ of oriented graphs such that an oriented graph $G$ has an injective oriented colouring with the given number of colours …

This paper presents tight bounds and characterizations for the vertex cover number and the connected vertex cover number of graphs. In particular, we identify all graphs for which βc(G) = |V (G)| − 1, proving that these are exactly the cycles and complete graphs. The analysis employs tools such as the degree matrix and the Rayleigh quotient to derive new and sharp upper bounds.

Let \(G=(V,E)\) be a connected, finite undirected graph. A set \(S \subseteq V\) is said to be a total dominating set of \(G\) if every vertex in \(V\) is adjacent to some vertex in \(S\). The total domination number, \(\gamma_{t}(G)\), is the minimum cardinality of a total dominating set in \(G\). We define the \(k\)-total bondage of $G$ to be the minimum number of edges to remove from \(G\) so …

For a planar graph G of order n, let F(G) be the set of all faces of G embedded into R 2 , including the exterior face. A bijective vertex labeling f : V (G) → {1, 2, ..., n} induces a face labeling f ∗ : F(G) → N defined by setting f ∗ (F) equal to the sum of all labels of the boundary vertices of F. The graph G is said to be hyper face-magic if there exists a vertex labeling whose induced face …

Is it known in the classical metric geometry literature which norms arise exactly as stable norms of periodic weighted graphs on $\mathbb{Z}^n$ and is there a known realization theorem?

Let $$S_n=\{(x,y)\in Z^2:x^2+y^2\leq n\}.$$ Construct a graph $G_n$ whose vertices are the points of $S_n$. Two vertices $u$ and $v$ are connected if and only if the line segment joining them contains ...

Let $A$ be a countably infinite graph, and $B$ a countably infinite graph which contains $A$ as an induced subgraph; let $\alpha$ be an automorphism of $A$; is there an embedding of $B$ in the Rado ...

Not THE famous ABC conjecture, but a conjecture of mine with Mohammed Aljohani and John Bamberg. A simple counting argument shows that, in any vertex-transitive graph, the product of the clique number and the independence number is at most the … Continue reading →

_Annals of Mathematics and Computer Science_ 34:1-12. 2026In this paper, we initiate the study of a new restricted parameter of convex Roman domination in graphs, called the outer-independent convex Roman dominating function, and discuss some of its combinatorial properties. ( direct link )

IntroductionWith the deep integration of generation-transmission-load-storage systems, the power demand side has become highly dynamic and stochastic, challenging the traditional assumption that user behavior remains stationary over time. Static clustering models therefore suffer from sensitivity to daily noise and false user identity switching.MethodsThis study proposes Dynamic Evolutionary Clus…

A finite, undirected graph $G=(V,E)$ is connected in the traditional sense if for all $v, w\in V$ with $v\neq w$ there is a finite path from $v$ to $w$. Moreover, $G$ is connected in the topological ...

Fix integers and and set . Let denote the complete -partite -uniform hypergraph with parts of size . We prove that the Zarankiewicz number provided . Previously this was known only for due to Pohoata and Zakharov. Our novel approach, which uses Behrend’s construction of sets with no 3-term arithmetic progression, also applies for small values of , for example, it gives where the exponent 11/4 is …

We call a hypergraph $H=(V,E)$ bipartite if there is $S\subseteq V$ with $e\cap S$ and $e\setminus S$ both nonempty for all $e\in E$. If $(\omega, E)$ is not bipartite and all members are infinite, do ...

Probability can become hard to reason about when many variables interact. One variable affects another. Evidence changes belief. Dependencies start to form a network. That is where Probabilistic Graphical Models become useful. Core Idea A Probabilistic Graphical Model represents uncertainty with a graph. The nodes are random variables. The edges represent relationships between them. Instead of tr…

This document develops a strict graph-theoretic formalization of the determination dependency graph G_D introduced in natural language in Structural Determination Theory (SDT) v5.9. Working at the level of partial-order theory, lattice theory, and elementary category theory, the formalization elevates SDT's core structural objects — nodes, edges, products, layers, traces — to a precisely defined …

I am taking a graph theory class and our professor started a new topic called graph labelling today, specifically, vertex labelling. Halfway through the lecture, I dozed off and missed the rest of it. ...

Bayesian Networks can feel confusing because they combine two things at once. Graphs show structure. Probabilities show uncertainty. The key is to see them as one model, not two separate topics. Core Idea A Bayesian Network represents relationships between variables using a directed graph. Each node is a variable. Each edge shows a dependency. Each node also has probability values that explain ho…