883 research outputs found
Location-domination in line graphs
A set of vertices of a graph is locating if every two distinct
vertices outside have distinct neighbors in ; that is, for distinct
vertices and outside , , where
denotes the open neighborhood of . If is also a dominating set (total
dominating set), it is called a locating-dominating set (respectively,
locating-total dominating set) of . A graph is twin-free if every two
distinct vertices of have distinct open and closed neighborhoods. It is
conjectured [D. Garijo, A. Gonzalez and A. Marquez, The difference between the
metric dimension and the determining number of a graph. Applied Mathematics and
Computation 249 (2014), 487--501] and [F. Foucaud and M. A. Henning.
Locating-total dominating sets in twin-free graphs: a conjecture. The
Electronic Journal of Combinatorics 23 (2016), P3.9] respectively, that any
twin-free graph without isolated vertices has a locating-dominating set of
size at most one-half its order and a locating-total dominating set of size at
most two-thirds its order. In this paper, we prove these two conjectures for
the class of line graphs. Both bounds are tight for this class, in the sense
that there are infinitely many connected line graphs for which equality holds
in the bounds.Comment: 23 pages, 2 figure
Circuits and Cycles in Graphs and Matroids
This dissertation mainly focuses on characterizing cycles and circuits in graphs, line graphs and matroids. We obtain the following advances.
1. Results in graphs and line graphs. For a connected graph G not isomorphic to a path, a cycle or a K1,3, let pc(G) denote the smallest integer n such that the nth iterated line graph Ln(G) is panconnected. A path P is a divalent path of G if the internal vertices of P are of degree 2 in G. If every edge of P is a cut edge of G, then P is a bridge divalent path of G; if the two ends of P are of degree s and t, respectively, then P is called a divalent (s, t)-path. Let l(G) = max{m : G has a divalent path of length m that is not both of length 2 and in a K3}. We prove the following. (i) If G is a connected triangular graph, then L(G) is panconnected if and only if G is essentially 3-edge-connected. (ii) pc(G) ≤ l(G) + 2. Furthermore, if l(G) ≥ 2, then pc(G) = l(G) + 2 if and only if for some integer t ≥ 3, G has a bridge divalent (3, t)-path of length l(G).
For a graph G, the supereulerian width μ′(G) of a graph G is the largest integer s such
that G has a spanning (k;u,v)-trail-system, for any integer k with 1 ≤ k ≤ s, and for any
u, v ∈ V (G) with u ̸= v. Thus μ′(G) ≥ 2 implies that G is supereulerian, and so graphs with
higher supereulerian width are natural generalizations of supereulerian graphs. Settling an open
problem of Bauer, Catlin in [J. Graph Theory 12 (1988), 29-45] proved that if a simple graph
G on n ≥ 17 vertices satisfy δ(G) ≥ n − 1, then μ′(G) ≥ 2. In this paper, we show that for 4
any real numbers a, b with 0 \u3c a \u3c 1 and any integer s \u3e 0, there exists a finite graph family
F = F(a,b,s) such that for a simple graph G with n = |V(G)|, if for any u,v ∈ V(G) with
uv ∈/ E(G), max{dG(u), dG(v)} ≥ an + b, then either μ′(G) ≥ s + 1 or G is contractible to a
member in F. When a = 1,b = −3, we show that if n is sufficiently large, K3,3 is the only 42
obstacle for a 3-edge-connected graph G to satisfy μ′(G) ≥ 3. An hourglass is a graph obtained from K5 by deleting the edges in a cycle of length 4, and an
hourglass-free graph is one that has no induced subgraph isomorphic to an hourglass. Kriesell in [J. Combin. Theory Ser. B, 82 (2001), 306-315] proved that every 4-connected hourglass-free line graph is Hamilton-connected, and Kaiser, Ryj ́aˇcek and Vr ́ana in [Discrete Mathematics, 321 (2014) 1-11] extended it by showing that every 4-connected hourglass-free line graph is 1- Hamilton-connected. We characterize all essentially 4-edge-connected graphs whose line graph is hourglass-free. Consequently we prove that for any integer s and for any hourglass-free line
graph L(G), each of the following holds. (i) If s ≥ 2, then L(G) is s-hamiltonian if and only if κ(L(G)) ≥ s + 2; (ii) If s ≥ 1, then L(G) is s-Hamilton-connected if and only if κ(L(G)) ≥ s + 3.
For integers s1, s2, s3 \u3e 0, let Ns1,s2,s3 denote the graph obtained by identifying each vertex of a K3 with an end vertex of three disjoint paths Ps1+1, Ps2+1, Ps3+1 of length s1,s2 and s3, respectively. We prove the following results. (i)LetN1 ={Ns1,s2,s3 :s1 \u3e0,s1 ≥s2 ≥s3 ≥0ands1+s2+s3 ≤6}. Thenforany N ∈ N1, every N-free line graph L(G) with |V (L(G))| ≥ s + 3 is s-hamiltonian if and only if κ(L(G)) ≥ s + 2. (ii)LetN2={Ns1,s2,s3 :s1\u3e0,s1≥s2≥s3≥0ands1+s2+s3≤4}.ThenforanyN∈N2, every N -free line graph L(G) with |V (L(G))| ≥ s + 3 is s-Hamilton-connected if and only if κ(L(G)) ≥ s + 3. 2. Results in matroids. A matroid M with a distinguished element e0 ∈ E(M) is a rooted matroid with e0 being the root. We present a characterization of all connected binary rooted matroids whose root lies in at most three circuits, and a characterization of all connected binary rooted matroids whose root lies in all but at most three circuits. While there exist infinitely many such matroids, the number of serial reductions of such matroids is finite. In particular, we find two finite families of binary matroids M1 and M2 and prove the following. (i) For some e0 ∈ E(M), M has at most three circuits containing e0 if and only if the serial reduction of M is isomorphic to a member in M1. (ii) If for some e0 ∈ E(M), M has at most three circuits not containing e0 if and only if the serial reduction of M is isomorphic to a member in M2. These characterizations will be applied to show that every connected binary matroid M with at least four circuits has a 1-hamiltonian circuit graph
Structural characterization of some problems on circle and interval graphs
A graph is circle if there is a family of chords in a circle such that two
vertices are adjacent if the corresponding chords cross each other. There are
diverse characterizations of circle graphs, many of them using the notions of
local complementation or split decomposition. However, there are no known
structural characterization by minimal forbidden induced subgraphs for circle
graphs. In this thesis, we give a characterization by forbidden induced
subgraphs of circle graphs within split graphs. A -matrix has the
consecutive-ones property (C1P) for the rows if there is a permutation of its
columns such that the 's in each row appear consecutively. In this thesis,
we develop characterizations by forbidden subconfigurations of -matrices
with the C1P for which the rows are -colorable under a certain adjacency
relationship, and we characterize structurally some auxiliary circle graph
subclasses that arise from these special matrices. Given a graph class , a
-completion of a graph is a graph such
that belongs to . A -completion of is minimal if does not belong to for every proper subset of . A
-completion of is minimum if for every -completion of , the cardinal of is less than or equal to the cardinal
of . In this thesis, we study the problem of completing minimally to obtain
a proper interval graph when the input is an interval graph. We find necessary
conditions that characterize a minimal completion in this particular case, and
we leave some conjectures for the future.Comment: PhD Thesis, joint supervision Universidad de Buenos
Aires-Universit\'e Paris-Nord. Dissertation took place on March 30th 202
A note on stability for maximal -free graphs
Popielarz, Sahasrabudhe and Snyder in 2018 proved that maximal -free
graphs with edges contain a
complete -partite subgraph on vertices. This was very recently
extended to odd cycles in place of by Wang, Wang, Yang and Yuan. We
further extend it to some other 3-chromatic graphs, and obtain some other
stability results along the way.Comment: 10 page
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