graph theory – What is Euler’s Formula when 2­-cell embedding condition was removed?

It is well known that Euler’s Formula for genus $gamma$: For every 2­-cell embedding of a graph
on a surface with genus $gamma$, the numbers of vertices, edges, and faces satisfy

My quetion: If the 2­-cell embedding condition was removed,what will we get?
Does following inequality hold true in general genus $gamma$?
$$2-2gammale n-e+fle 2-2(gamma-1)=2 gamma$$
For example: We know a toroidal graph is a graph that can be embedded on a torus. So maybe for embedding of any toroidal graph, we would get
$$0le n-e+fle 2. $$

If we consider following graph $C_6+e$, and the toroidal embedding of $C_6+e$ contains $6$ vertices , $7$ edges and $2$ faces. so $n-e+f=6-7+2=1$.
Obviously $0le 1 le 2$.
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Graph API Delta Query tracking

Is there any option to track the events in MS Graph API at tenant level using Delta Query? I have a requirement of capture all the Teams Events with BraodcastMeeting equal to true (means live event in Teams). Can anybody please help me on this?

dg.differential geometry – How to make a cusp catastrophes’ graph model in 3D?

I have a problem of finding a way or at least an example how to make a catastrophe model in 3D. I have googled and searched in books, but all I have is a description with no examples, I have only a formula which does not work and show the catastrophe in 3D in GeoGebra.

I would be more than thankful if someone helped me!

Thank you in advance.

colorings – Triangles covering all vertices of a tri-partite graph

This question is an extension of this one: Min path cover for a three-layer graph with all paths traversing all layers.

I’m designing fictional fruits. Each fruit has three attributes; color, taste and smell. Also, each of the values of the attributes have some compatibility with the values of the other attributes. So, we get a tri-partite graph. An example is shown below. Here, 1 and 2 are color attributes, 3,4 and 5 are taste attributes and 6 and 7 are smell attributes. Also, color-1 is compatible with taste-3 and smell-6 and so on.

Fig 1: fruit graph

I want to design a minimal number of fruits while still covering all attributes (all 2 colors, all 3 tastes and all 2 smells in this case). For example, the example graph above has the following solution (shown with pink lines, fruit-1 has color-1, taste-3 and smell-6; fruit-2 has color-1, taste-4 and smell-7 while fruit-3 has color-2, taste-5 and smell-6; hence covering all levels of all attributes with 3 fruits):

enter image description here

We know this is optimal since there are three tastes and we couldn’t have used less than 3 fruits in this case.

The question is: how to design an algorithm to get the minimum number of fruits required and their configurations given a general graph like the one specified above.

Note that it might not have been possible to cover all attributes even if the graph has no isolated vertices and I asked a question on feasibility here:

graph theory – Increasing the Hadwiger number by making any pair of non-adjacent points adjacent

Let $G=(V,E)$ be a finite, simple, undirected graph. The Hadwiger number $eta(G)$ of $G$ is defined to be the largest positive integer $ninmathbb{N}$ such that the complete graph $K_n$ is a minor of $G$.

We say that a graph is vertex-critical if removing any vertex reduces the chromatic number.

Question. What is an example of a connected, non-complete vertex-critical graph $G=(V,E)$ with the following property?

Adding any edge increases the Hadwiger number — or more formally: for $vneq w in V$ with ${v,w}notin E$ we get $$etabig((V,Ecupbig{{v,w}big})big) > etabig((V,E)big).$$

Motivation. Any minimal counterexample to Hadwiger’s conjecture has the properties given in the question.

A graph database suitable for analyzing a heap snapshot?

It looks like recommendation questions aren’t explicitly OT, so here goes:

I’m working on some tooling for analyzing a dump of the heap of a running program. The dump is just a list of nodes with associated metadata and references to other nodes (possibly-cyclical).

I don’t have any experience with graph databases, and I’m wondering if I would save myself a lot of time by building tooling around a graph DB. So I’m looking for recommendations and pointers to resources, and advice.

Some specific questions:

  • are there any graph databases that have functionality built in for computing a dominator tree? (googling this didn’t seem to get any results)
  • are there any DBs that have tooling for visualizing a huge graph?

graph theory – Generating triangulations with given topology

I am looking for information about the problem of identifying the heaviest minimal subset $Fsubset E$ of the edgeset $E$ of a complete symmetric graph $G(V,E)$ with randomly weighted edges such that $Gsetminus F$ has a given topology, e.g. is planar, and a triangulation, i.e. every remaining edge is an edge of a 3-cycle.

An exemplary problem would be to find a triangulation of a complete graph, whose vertices resemble points on a torus and edge weigths equal to euclidean distance, that allows for a planar embedding.

Being able to calculate the lightest planar triangulation of graphs with arbitrarily weighted edges would e.g. yield a meaningful definition of the planar convex hull of such graphs and thus yield an initial tour that can be expanded to a lightest Hamilton cycle by successive integration of vertices.

algorithms – Optimization problem over bidirectional connected graph

A company has several automatic vertical warehouses (called elevators). Each elevator have several trays and each tray has several slots. A slot contains a given quantity of a given article. Elevators, trays and slots are identified by unique IDs and slot also have access to the ID of their own article. In the same tray there can be multiple slots with the same article.

I have to design an algorithm which, given an order list (we can model it as a dictionary with articles’ IDs as key and quantity needed as values), returns the minimum list of trays needed to satisfy the order.

This is quite different from a standard warehouse optimization problem because we are not considering the physical distances between each tray, since the elevators are automatic and they give us the tray, while in the classical problem is the human who moves toward the tray to pick the item from it.

A distance function is given: d(a, b) which returns 1 if tray a and b are on different elevators and returns 2 if they’re on the same one. That could be counter-intuitive, but remember that these are vertical elevators, so the time needed to change tray on the same elevator is greater than the time to move to a different elevator with the tray already in bay. Furthermore, if we use more elevators at the same time, we can “parallelize” the picking process (man A pick from elevator 1 while man B picks from elevator 2…) Anyway, this function is given and I cannot change it.

After reading through articles about warehouse optimization, I’ve decided to take an heuristic approach.
I’ve modeled a sort of euclidean space, with an axes for each article contained in the order. Then we can consider a tray as a point on that space, with coordinates for each axes equal to the quantity that tray has of the article corresponding to that axes. In the same way we can imagine our order as a point.
Then I’ve create a heuristic function, f(order o, tray t), which returns the “euclidean distance” from point-tray t to the point-order o. The idea is that the more a tray is “near” the order, the more article we can pick out of it.

So, to satisfy the order, I simply compute f(order o, tray t) for each tray in every elevator. Then I order it by descending value and finally I greedily take a tray with the minimum distance from the order. This will be repeated until we collect enough articles to satisfy the order.

Now I `d like to find a better solution, taking into account also the physical distances from tray returned by function d.

I`ve tried to build a graph in which each node is a tray and is connected to each of the others by a directed edge. The weights of the edge from node i to node j will be equal to the physical distance from tray i to tray j, plus the heuristic function computed on tray j

--> w(i, j) = d(i, j) + f(order, j)

This results in a fully-connected, bidirectional graph (each node linked with each other node by both an incoming and an outgoing edge).
I want to apply some algorithm of shortest path (or any other useful algorithm) taken from graph theory on this graph but I couldnt find anything really helpful. Ive tried to apply the A* search algorithm (using function d as gScore and my function f as heuristic) but it gives me no result. I think A* can`t be applied in such a graph (bidirectional and fully connected).

Is there any algorithm I can apply on such a graph? Or maybe the graph is not the right structure to represent my problem. I`m open to new solutions.

co.combinatorics – The density of a tripartite 1-planar graph

1-planar graphs are those can be drawn in the plane so that there is at most one crossing per edge. We know that the density of an $n$-vertex 1-planar graph is at most $4n-8$, and the density of a bipartite $n$-vertex 1-planar graph is at most $3n-6$. I remember that there is also a result on the the density of a tripartite $n$-vertex 1-planar graph but I can’t find the sourse now. Does anyone know this result or the sourse? Thanks in advance!

sharepoint online – createLink graph api request

I am trying to create a sharedURL via a picker in my app that constucts a web url with a query par linked to a sharepoint file, which reads into the app on load. For some reason if I grab the manual sharedURL from sharepoint it works, as I endode it and point to:

const url =!${encodedString}/driveItem;

With auth and grabbing the @microsoft.graph.downloadUrl I can easily read into memory and into my app. so all good!

However I cannnot get the createLink from graph api to work as it does not include the ?e= query par, like a manual share option does. It would be good to know what it represents as I cannot access the content without it.

const createLink =${driveId}/items/${itemId}/createLink;

I get back a successful 200 or 201 on POST the create the link and has the same weburl but without ?e= so am thinking that is the issue. but when I read it in, it says Requested sharing link could not be found.

Could someone please advise pls?