#BellmanFord: unlike #Dijkstra's, it can handle negative arcs. But #SSNC isn't just about any paths. It's about paths of length 1 & 2 (1st & 2nd neighborhoods). Does Bellman-Ford address this? #SeymourConjecture #Research #OpenProblem #Math #ComputerScience

#SeymourConjecture: In its standard form, #Dijkstra's algorithm is ill-equipped to address the #SSNC's core challenge: analyzing the size of 1st & 2nd neighborhoods in #OrientedGraphs. What modifications could make it more relevant? #TCS #ComputerScience

#SeymourConjecture: #Dijkstra's finds #ShortestPaths, crucial for general graph analysis. However, SSNC requires analyzing all paths of length 1 & 2, not just the shortest. Negative arcs/back arcs also complicate this.

Day 19 of #SSNC facts! We're exploring why the #SeymourConjecture remains unsolved. Today, we examine how #algorithms like #Dijkstra #Algorithm handle the challenge of neighborhood analysis. #math #mathematics #compSci #computerScience

Day 18 of #SSNC facts! We're looking at the #SeymourConjecture: Does every #OrientedGraph have a node with a second neighborhood at least as large as its first? Can traditional #GraphTheory #Algorithms solve the neighborhood size challenge? Today, we examine #DepthFirstSearch (DFS). #Math

Let's illustrate: Consider this #OrientedGraph: x->y->z, y->w, z->w. BFS assigns levels correctly, but the back arc (z->w) creates a path that distorts the second neighborhood of 'y' for #SSNC analysis. Back arcs are a problem for the #SeymourConjecture and must be handled.

Yesterday we explored BFS: a general traversal #algorithm, not a #SeymourConjecture solver. It's designed for level-order exploration, not neighborhood analysis. This means it doesn't inherently handle back arcs, leading to potential misrepresentations of neighborhoods in #OrientedGraphs. #math

Day 17 of #SSNC facts! Remember, we're exploring: Does every #OrientedGraph have a node with a second neighborhood at least as large as its first? Why is this deceptively simple question still unsolved? #GraphTheory #Resesrch #Math #ComputerScience #SeymourConjecture

Day 16 of #SSNC facts! We've been discussing the #SeymourConjecture: It asks the questionv Does every #OrientedGraph have a node whose second neighborhood is at least as large as its first? Open since 1990, but why is this still unsolved? Let's explore.

#GraphTheory #Math #ComputerScience

Day 15 of #SSNC facts. Remember, for a Constraint Satisfaction Problem, we're not looking for a solution, but proving there ISN'T one. If a CSP is 'unsatisfiable', it means certain scenarios that break the #SSNC are impossible, improving our bounds! #MathLogic #Bounds #SeymourConjecture

How do we turn a graph problem into a CSP? We translate the relationships between vertices and their neighbors into mathematical constraints. This is the 'rule' set. #SeymourConjecture #GraphTheory #mathematics

Day 14 of #SSNC facts! We're diving into Constraint Satisfaction Problems (CSPs). Let's talk about how they are used to tackle the #SeymourConjecture. Think of CSPs as a set of rules we ask a computer to try and solve. #GraphTheory #CSP #math

Day 13 of #SSNC facts! Did you know #DominatingSets are relevant to the #SeymourConjecture? A dominating set is a vertex subset where every other vertex is adjacent to one in the set. How might this relate to neighborhoods? #GraphTheory #SSNC #math #compsci #ComputerScience

#Challenge! 🧠 Given the attached #OrientedGraph, what node order maximizes the number of forward edges? (Median Order!) Havet & Thomassé used this concept in their work. Show your solution! #GraphTheory #MedianOrder #Algorithms #TournamentTheory #DeanConjecture #SeymourConjecture #math #mathematics

Day 12 of #SSNC facts! Fisher: Existence proof (probabilistic). Thomassé: Constructive proof (median orders). Both solved Dean's Conjecture, but Thomassé provided a path to #algorithms. This was a fundamental shift in approach! #MathematicalProofs #AlgorithmDesign #GraphTheory #SeymourConjecture

Thomassé's use of median orders showcases the power of combinatorial arguments in #GraphTheory. It provides a more direct and intuitive understanding of the underlying structure of tournaments. #CombinatorialMath #GraphStructure #Mathematics #DeanConjecture #SeymourConjecture #math

Unlike Fisher's probabilistic existence proof, Thomassé's method is constructive, #algorithmic. It provides a way to find the vertex satisfying the #DeanConjecture. This is a game-changer for applications! #Algorithms #ConstructiveProof #TournamentTheory #SeymourConjecture

Day 10 of #SSNC facts! The #DeanConjecture states in any tournament, there's a vertex 'v' where |out-neighborhood(v, 2)| ≥ |out-neighborhood(v, 1)|. Simply put, you can reach more vertices in 2 steps than 1. Fisher provided a probabilistic proof. #GraphTheory #SeymourConjecture #math #research

This version of the problem was proposed by Dean and Latka. It is a special case of the #SeymourConjecture, restricted to tournaments. A special note is that this is where the general Seymour Conjecture was first published in print #math #research #SSNC

Open problems like #SeymourConjecture or #SullivanConjecture teach persistence. The ability to keep exploring, even when faced with setbacks, is a skill that can be applied to any challenge. What skills have you gained from tackling challenging problems? #Persistence #Resilience #NeverGiveUp

While related to the well-known #SeymourConjecture (proposed ~1990/95), the #SullivanConjecture is more recent (2006). Consequently, there's comparatively less research on this second neighborhood problem.
#GraphTheory #ComputerScience #OpenProblems #ResearchSkills #Math

Day 8 of SSNC facts! Yesterday we mentioned that the #SeymourConjecture isn't alone. Lets look at the #SullivanConjecture. It asks if the out-degrees of a node's second out-neighborhood are at least twice its in-degree. How are these conjectures similar? Different? #GraphTheory #OpenProblems #math

It's not just related conjectures! Tackling #OpenProblems like #SeymourConjecture reinforces classroom #Algorithms like #BFS and #DFS. Practical application deepens understanding. #GraphTheory #ComputerScience #Math #Dev #Programming

Sometimes, progress on one conjecture can shed light on others. The tools and techniques developed to study the #SeymourConjecture might also be useful for tackling the #SullivanConjecture. #Interconnectedness #Research

Day 7 of the #SSNC facts! The #SeymourConjecture isn't the only conjecture in graph theory. There are related conjectures, like the #SullivanConjecture and the #DeanConjecture, that explore similar ideas about graph structure and neighborhoods. #GraphTheory #OpenProblems

This is just a simple statement But it's also a decades-long puzzle? Is the #SeymourConjecture like an unbeatable video game challenge? What hidden complexity lies in these #OrientedGraphs? #MathProblems #Unsolved #GraphTheory #OpenProblems

In these fields that can be represented by #OrientedGraphs (transportation, social media, #KnowledgeGraphs and more), a solution to the #SeymourConjecture helps is locate the "influencers", the ones who being are talked about. #math #GraphTheory #ComputerScience

Day 6 of #SSNC facts! Now that we've covered the basic definitions, let's talk about why the #SeymourConjecture (which asks if an #OrientedGraph has a node whose second neighborhood is at least as large as the first). #math #GraphTheory #ComputerScience

Here's a #programming question. How could we distinguish between a node's first and second neighbor's in an #OrientedGraph? What problems could arise? #SeymourConjecture #GraphTheory #Developers #dev #TCS #Coding #computerscience

The #SeymourConjecture states that there exists a vertex in an #OrientedGraph has at least as many vertices in its first second out-neighborhood as in its first out-neighborhood. This Conjecture has remained open since 1990. #OpenProblem #SSNC #GraphTheory #math #mathematics #mathsky #compsky