Standard term for “dense” subset of a graph
$begingroup$
Let us temporarily say that a subset $A$ of (the set of vertices of) an undirected graph $G$ is dense when each vertex of $G$ is adjacent to an element of $A$. In other words, the (distance-$1$, open) neighborhood of $A$ is all of $G$.
Question 0: I suppose there is a more standard term for this than "dense", which I just made up. What is it?
Now let us consider the following invariant $psi(G)$ of an undirected graph $G$: it is the largest number $n$ of pairwise disjoint dense subsets of $G$ (where "dense" is defined as above). I.e., it is the largest $n$ such that we can find $A_1,ldots,A_n$ pairwise disjoint subsets of $G$ such that every vertex of $G$ is a adjacent to an element of each one of the $A_i$. (We agree that $psi(G)=0$ if no dense subset exists, i.e., whenever some vertex is isolated.)
Evidently, $psi(G)$ is at most the minimal degree $delta(G)$ of $G$.
Question 1: Does $psi(G)$ have a standard name and/or notation? If so, what is it?
Question 2: Is there some literature pertaining to this quantity?
Motivation: Consider the information theory protocol where Alice is given a vertex $x$ of $G$ and must choose a neighbor $x'$ of $x$ which she will communicate to her accomplice Bob (Bob is not given the value of $x$); Alice is trying to communicate some information to Bob: the obvious protocol (namely, Bob receives the information $i$ where $x'in A_i$) will allow Alice to pass $log n$ bits of information to Bob where $n = psi(G)$. See here for a motivation of the motivation.
Edit: As Michal Adamaszek answers in the comments, the answer to question 0 is "total dominating" (in contrast, a "dominating" set $A$ is one such that every vertex of $G$ is at distance $leq 1$ to a vertex in $A$). Now knowing this, Google returns a concept closely related to my $psi(G)$, namely the domatic number of a graph, which is the maximum number of pairwise disjoint dominating sets. It therefore seems logical to call my $psi(G)$ the "total domatic number", and some authors seem to have done so.
combinatorics graph-theory
asked Dec 12 '18 at 12:58
Gro-TsenGro-Tsen
3,9221323
$endgroup$
add a comment |
$begingroup$
Let us temporarily say that a subset $A$ of (the set of vertices of) an undirected graph $G$ is dense when each vertex of $G$ is adjacent to an element of $A$. In other words, the (distance-$1$, open) neighborhood of $A$ is all of $G$.
Question 0: I suppose there is a more standard term for this than "dense", which I just made up. What is it?
Now let us consider the following invariant $psi(G)$ of an undirected graph $G$: it is the largest number $n$ of pairwise disjoint dense subsets of $G$ (where "dense" is defined as above). I.e., it is the largest $n$ such that we can find $A_1,ldots,A_n$ pairwise disjoint subsets of $G$ such that every vertex of $G$ is a adjacent to an element of each one of the $A_i$. (We agree that $psi(G)=0$ if no dense subset exists, i.e., whenever some vertex is isolated.)
Evidently, $psi(G)$ is at most the minimal degree $delta(G)$ of $G$.
Question 1: Does $psi(G)$ have a standard name and/or notation? If so, what is it?
Question 2: Is there some literature pertaining to this quantity?
Motivation: Consider the information theory protocol where Alice is given a vertex $x$ of $G$ and must choose a neighbor $x'$ of $x$ which she will communicate to her accomplice Bob (Bob is not given the value of $x$); Alice is trying to communicate some information to Bob: the obvious protocol (namely, Bob receives the information $i$ where $x'in A_i$) will allow Alice to pass $log n$ bits of information to Bob where $n = psi(G)$. See here for a motivation of the motivation.
Edit: As Michal Adamaszek answers in the comments, the answer to question 0 is "total dominating" (in contrast, a "dominating" set $A$ is one such that every vertex of $G$ is at distance $leq 1$ to a vertex in $A$). Now knowing this, Google returns a concept closely related to my $psi(G)$, namely the domatic number of a graph, which is the maximum number of pairwise disjoint dominating sets. It therefore seems logical to call my $psi(G)$ the "total domatic number", and some authors seem to have done so.
combinatorics graph-theory
asked Dec 12 '18 at 12:58
Gro-TsenGro-Tsen
3,9221323
$endgroup$
2
$begingroup$
What you call dense is called a "total dominating set in a graph".
$endgroup$
– Michal Adamaszek
Dec 12 '18 at 13:19
$begingroup$
Why say "distance-1 open" ( just wondering about use of term open here) ?
$endgroup$
– coffeemath
Dec 12 '18 at 13:22
1
$begingroup$
@coffeemath I'm not too familiar with graph-theoretic terminology (which is rather at odd with terminology of other domains I'm more accustomed to), but I understand that the "closed" neighborhood of a vertex includes said vertex whereas the "open" nhbd does not.
$endgroup$
– Gro-Tsen
Dec 12 '18 at 13:56
$begingroup$
@MichalAdamaszek Thank you. This answers my question 0 and allows me to google answers very close to my questions 1 and 2: the term "domatic number" seems very close to my $psi(G)$.
$endgroup$
– Gro-Tsen
Dec 12 '18 at 14:02
add a comment |
$begingroup$
Let us temporarily say that a subset $A$ of (the set of vertices of) an undirected graph $G$ is dense when each vertex of $G$ is adjacent to an element of $A$. In other words, the (distance-$1$, open) neighborhood of $A$ is all of $G$.
Question 0: I suppose there is a more standard term for this than "dense", which I just made up. What is it?
Now let us consider the following invariant $psi(G)$ of an undirected graph $G$: it is the largest number $n$ of pairwise disjoint dense subsets of $G$ (where "dense" is defined as above). I.e., it is the largest $n$ such that we can find $A_1,ldots,A_n$ pairwise disjoint subsets of $G$ such that every vertex of $G$ is a adjacent to an element of each one of the $A_i$. (We agree that $psi(G)=0$ if no dense subset exists, i.e., whenever some vertex is isolated.)
Evidently, $psi(G)$ is at most the minimal degree $delta(G)$ of $G$.
Question 1: Does $psi(G)$ have a standard name and/or notation? If so, what is it?
Question 2: Is there some literature pertaining to this quantity?
Motivation: Consider the information theory protocol where Alice is given a vertex $x$ of $G$ and must choose a neighbor $x'$ of $x$ which she will communicate to her accomplice Bob (Bob is not given the value of $x$); Alice is trying to communicate some information to Bob: the obvious protocol (namely, Bob receives the information $i$ where $x'in A_i$) will allow Alice to pass $log n$ bits of information to Bob where $n = psi(G)$. See here for a motivation of the motivation.
Edit: As Michal Adamaszek answers in the comments, the answer to question 0 is "total dominating" (in contrast, a "dominating" set $A$ is one such that every vertex of $G$ is at distance $leq 1$ to a vertex in $A$). Now knowing this, Google returns a concept closely related to my $psi(G)$, namely the domatic number of a graph, which is the maximum number of pairwise disjoint dominating sets. It therefore seems logical to call my $psi(G)$ the "total domatic number", and some authors seem to have done so.
combinatorics graph-theory
asked Dec 12 '18 at 12:58
Gro-TsenGro-Tsen
3,9221323
$endgroup$
Let us temporarily say that a subset $A$ of (the set of vertices of) an undirected graph $G$ is dense when each vertex of $G$ is adjacent to an element of $A$. In other words, the (distance-$1$, open) neighborhood of $A$ is all of $G$.
Question 0: I suppose there is a more standard term for this than "dense", which I just made up. What is it?
Now let us consider the following invariant $psi(G)$ of an undirected graph $G$: it is the largest number $n$ of pairwise disjoint dense subsets of $G$ (where "dense" is defined as above). I.e., it is the largest $n$ such that we can find $A_1,ldots,A_n$ pairwise disjoint subsets of $G$ such that every vertex of $G$ is a adjacent to an element of each one of the $A_i$. (We agree that $psi(G)=0$ if no dense subset exists, i.e., whenever some vertex is isolated.)
Evidently, $psi(G)$ is at most the minimal degree $delta(G)$ of $G$.
Question 1: Does $psi(G)$ have a standard name and/or notation? If so, what is it?
Question 2: Is there some literature pertaining to this quantity?
Motivation: Consider the information theory protocol where Alice is given a vertex $x$ of $G$ and must choose a neighbor $x'$ of $x$ which she will communicate to her accomplice Bob (Bob is not given the value of $x$); Alice is trying to communicate some information to Bob: the obvious protocol (namely, Bob receives the information $i$ where $x'in A_i$) will allow Alice to pass $log n$ bits of information to Bob where $n = psi(G)$. See here for a motivation of the motivation.
Edit: As Michal Adamaszek answers in the comments, the answer to question 0 is "total dominating" (in contrast, a "dominating" set $A$ is one such that every vertex of $G$ is at distance $leq 1$ to a vertex in $A$). Now knowing this, Google returns a concept closely related to my $psi(G)$, namely the domatic number of a graph, which is the maximum number of pairwise disjoint dominating sets. It therefore seems logical to call my $psi(G)$ the "total domatic number", and some authors seem to have done so.
combinatorics graph-theory
combinatorics graph-theory
asked Dec 12 '18 at 12:58
Gro-TsenGro-Tsen
3,9221323
asked Dec 12 '18 at 12:58
Gro-TsenGro-Tsen
3,9221323
edited Dec 12 '18 at 14:07
Gro-Tsen
asked Dec 12 '18 at 12:58
Gro-TsenGro-Tsen
3,9221323
asked Dec 12 '18 at 12:58
Gro-TsenGro-Tsen
3,9221323
asked Dec 12 '18 at 12:58
Gro-TsenGro-Tsen
3,9221323
3,9221323
2
$begingroup$
What you call dense is called a "total dominating set in a graph".
$endgroup$
– Michal Adamaszek
Dec 12 '18 at 13:19
$begingroup$
Why say "distance-1 open" ( just wondering about use of term open here) ?
$endgroup$
– coffeemath
Dec 12 '18 at 13:22
1
$begingroup$
@coffeemath I'm not too familiar with graph-theoretic terminology (which is rather at odd with terminology of other domains I'm more accustomed to), but I understand that the "closed" neighborhood of a vertex includes said vertex whereas the "open" nhbd does not.
$endgroup$
– Gro-Tsen
Dec 12 '18 at 13:56
$begingroup$
@MichalAdamaszek Thank you. This answers my question 0 and allows me to google answers very close to my questions 1 and 2: the term "domatic number" seems very close to my $psi(G)$.
$endgroup$
– Gro-Tsen
Dec 12 '18 at 14:02
add a comment |
2
$begingroup$
What you call dense is called a "total dominating set in a graph".
$endgroup$
– Michal Adamaszek
Dec 12 '18 at 13:19
$begingroup$
Why say "distance-1 open" ( just wondering about use of term open here) ?
$endgroup$
– coffeemath
Dec 12 '18 at 13:22
1
$begingroup$
@coffeemath I'm not too familiar with graph-theoretic terminology (which is rather at odd with terminology of other domains I'm more accustomed to), but I understand that the "closed" neighborhood of a vertex includes said vertex whereas the "open" nhbd does not.
$endgroup$
– Gro-Tsen
Dec 12 '18 at 13:56
$begingroup$
@MichalAdamaszek Thank you. This answers my question 0 and allows me to google answers very close to my questions 1 and 2: the term "domatic number" seems very close to my $psi(G)$.
$endgroup$
– Gro-Tsen
Dec 12 '18 at 14:02
2
2
$begingroup$
What you call dense is called a "total dominating set in a graph".
$endgroup$
– Michal Adamaszek
Dec 12 '18 at 13:19
$begingroup$
What you call dense is called a "total dominating set in a graph".
$endgroup$
– Michal Adamaszek
Dec 12 '18 at 13:19
$begingroup$
Why say "distance-1 open" ( just wondering about use of term open here) ?
$endgroup$
– coffeemath
Dec 12 '18 at 13:22
$begingroup$
Why say "distance-1 open" ( just wondering about use of term open here) ?
$endgroup$
– coffeemath
Dec 12 '18 at 13:22
1
1
$begingroup$
@coffeemath I'm not too familiar with graph-theoretic terminology (which is rather at odd with terminology of other domains I'm more accustomed to), but I understand that the "closed" neighborhood of a vertex includes said vertex whereas the "open" nhbd does not.
$endgroup$
– Gro-Tsen
Dec 12 '18 at 13:56
$begingroup$
@coffeemath I'm not too familiar with graph-theoretic terminology (which is rather at odd with terminology of other domains I'm more accustomed to), but I understand that the "closed" neighborhood of a vertex includes said vertex whereas the "open" nhbd does not.
$endgroup$
– Gro-Tsen
Dec 12 '18 at 13:56
$begingroup$
@MichalAdamaszek Thank you. This answers my question 0 and allows me to google answers very close to my questions 1 and 2: the term "domatic number" seems very close to my $psi(G)$.
$endgroup$
– Gro-Tsen
Dec 12 '18 at 14:02
$begingroup$
@MichalAdamaszek Thank you. This answers my question 0 and allows me to google answers very close to my questions 1 and 2: the term "domatic number" seems very close to my $psi(G)$.
$endgroup$
– Gro-Tsen
Dec 12 '18 at 14:02
add a comment |
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$begingroup$
What you call dense is called a "total dominating set in a graph".
$endgroup$
– Michal Adamaszek
Dec 12 '18 at 13:19
$begingroup$
Why say "distance-1 open" ( just wondering about use of term open here) ?
$endgroup$
– coffeemath
Dec 12 '18 at 13:22
1
$begingroup$
@coffeemath I'm not too familiar with graph-theoretic terminology (which is rather at odd with terminology of other domains I'm more accustomed to), but I understand that the "closed" neighborhood of a vertex includes said vertex whereas the "open" nhbd does not.
$endgroup$
– Gro-Tsen
Dec 12 '18 at 13:56
$begingroup$
@MichalAdamaszek Thank you. This answers my question 0 and allows me to google answers very close to my questions 1 and 2: the term "domatic number" seems very close to my $psi(G)$.
$endgroup$
– Gro-Tsen
Dec 12 '18 at 14:02