scholarly journals Matrix Partitions with Finitely Many Obstructions

10.37236/976 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Tomás Feder ◽  
Pavol Hell ◽  
Wing Xie
Keyword(s):  
Symmetric Matrix ◽  
Forbidden Subgraph ◽  
Input Graph ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Forbidden Induced Subgraphs ◽  
Extra Effort ◽  
Matrix Partition ◽  
Finite Set ◽  
Forbidden Induced Subgraph

Each $m$ by $m$ symmetric matrix $M$ over $0, 1, *$, defines a partition problem, in which an input graph $G$ is to be partitioned into $m$ parts with adjacencies governed by $M$, in the sense that two distinct vertices in (possibly equal) parts $i$ and $j$ are adjacent if $M(i,j)=1$, and nonadjacent if $M(i,j)=0$. (The entry $*$ implies no restriction.) We ask which matrix partition problems admit a characterization by a finite set of forbidden induced subgraphs. We prove that matrices containing a certain two by two diagonal submatrix $S$ never have such characterizations. We then develop a recursive technique that allows us (with some extra effort) to verify that matrices without $S$ of size five or less always have a finite forbidden induced subgraph characterization. However, we exhibit a six by six matrix without $S$ which cannot be characterized by finitely many induced subgraphs. We also explore the connection between finite forbidden subgraph characterizations and related questions on the descriptive and computational complexity of matrix partition problems.

scholarly journals On Betti numbers of flag complexes with forbidden induced subgraphs

2019 ◽  
Vol 168 (3) ◽  
pp. 567-600
Author(s):  
KARIM ADIPRASITO ◽  
ERAN NEVO ◽  
MARTIN TANCER
Keyword(s):  
Growth Rate ◽  
Betti Number ◽  
Betti Numbers ◽  
Interesting Case ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Theorem Of The Alternative ◽  
Forbidden Induced Subgraphs ◽  
Forbidden Induced Subgraph ◽  
Flag Complexes

AbstractWe analyse the asymptotic extremal growth rate of the Betti numbers of clique complexes of graphs on n vertices not containing a fixed forbidden induced subgraph H.In particular, we prove a theorem of the alternative: for any H the growth rate achieves exactly one of five possible exponentials, that is, independent of the field of coefficients, the nth root of the maximal total Betti number over n-vertex graphs with no induced copy of H has a limit, as n tends to infinity, and, ranging over all H, exactly five different limits are attained.For the interesting case where H is the 4-cycle, the above limit is 1, and we prove a superpolynomial upper bound.

scholarly journals Toughness, Forbidden Subgraphs and Pancyclicity

2021 ◽  
Vol 37 (3) ◽  
pp. 839-866
Author(s):  
Wei Zheng ◽  
Hajo Broersma ◽  
Ligong Wang
Keyword(s):  
Recent Article ◽  
Forbidden Subgraph ◽  
Forbidden Subgraphs ◽  
Free Graph ◽  
Induced Subgraph ◽  
Complete Answer ◽  
Open Case

AbstractMotivated by several conjectures due to Nikoghosyan, in a recent article due to Li et al., the aim was to characterize all possible graphs H such that every 1-tough H-free graph is hamiltonian. The almost complete answer was given there by the conclusion that every proper induced subgraph H of $$K_1\cup P_4$$ K 1 ∪ P 4 can act as a forbidden subgraph to ensure that every 1-tough H-free graph is hamiltonian, and that there is no other forbidden subgraph with this property, except possibly for the graph $$K_1\cup P_4$$ K 1 ∪ P 4 itself. The hamiltonicity of 1-tough $$K_1\cup P_4$$ K 1 ∪ P 4 -free graphs, as conjectured by Nikoghosyan, was left there as an open case. In this paper, we consider the stronger property of pancyclicity under the same condition. We find that the results are completely analogous to the hamiltonian case: every graph H such that any 1-tough H-free graph is hamiltonian also ensures that every 1-tough H-free graph is pancyclic, except for a few specific classes of graphs. Moreover, there is no other forbidden subgraph having this property. With respect to the open case for hamiltonicity of 1-tough $$K_1\cup P_4$$ K 1 ∪ P 4 -free graphs we give infinite families of graphs that are not pancyclic.

scholarly journals Classes of graphs with restricted interval models

1999 ◽  
Vol Vol. 3 no. 4 ◽  
Author(s):  
Andrzej Proskurowski ◽  
Jan Arne Telle
Keyword(s):  
Maximum Clique ◽  
Interval Graphs ◽  
Induced Subgraph ◽  
Graph Theoretic ◽  
Graph Classes ◽  
International Audience ◽  
Forbidden Induced Subgraph ◽  
Nested Hierarchy ◽  
Graph Families

International audience We introduce q-proper interval graphs as interval graphs with interval models in which no interval is properly contained in more than q other intervals, and also provide a forbidden induced subgraph characterization of this class of graphs. We initiate a graph-theoretic study of subgraphs of q-proper interval graphs with maximum clique size k+1 and give an equivalent characterization of these graphs by restricted path-decomposition. By allowing the parameter q to vary from 0 to k, we obtain a nested hierarchy of graph families, from graphs of bandwidth at most k to graphs of pathwidth at most k. Allowing both parameters to vary, we have an infinite lattice of graph classes ordered by containment.

scholarly journals A forbidden induced subgraph characterization of distance-hereditary 5-leaf powers

2009 ◽  
Vol 309 (12) ◽  
pp. 3843-3852 ◽  
Author(s):  
Andreas Brandstädt ◽  
Van Bang Le ◽  
Dieter Rautenbach
Keyword(s):  
Induced Subgraph ◽  
Forbidden Induced Subgraph

INDUCED SUBGRAPHS OF GAMMA GRAPHS

2013 ◽  
Vol 05 (03) ◽  
pp. 1350012 ◽  
Author(s):  
N. SRIDHARAN ◽  
S. AMUTHA ◽  
S. B. RAO
Keyword(s):  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Vertex Set ◽  
Isomorphic Graphs

Let G be a graph. The gamma graph of G denoted by γ ⋅ G is the graph with vertex set V(γ ⋅ G) as the set of all γ-sets of G and two vertices D and S of γ ⋅ G are adjacent if and only if |D ∩ S| = γ(G) – 1. A graph H is said to be a γ-graph if there exists a graph G such that γ ⋅ G is isomorphic to H. In this paper, we show that every induced subgraph of a γ-graph is also a γ-graph. Furthermore, if we prove that H is a γ-graph, then there exists a sequence {Gn} of non-isomorphic graphs such that H = γ ⋅ Gn for every n.

scholarly journals Forbidden induced subgraphs for star-free graphs

2011 ◽  
Vol 311 (21) ◽  
pp. 2475-2484 ◽  
Author(s):  
Jun Fujisawa ◽  
Katsuhiro Ota ◽  
Kenta Ozeki ◽  
Gabriel Sueiro
Keyword(s):  
Induced Subgraphs ◽  
Forbidden Induced Subgraphs

scholarly journals Large Induced Subgraphs with All Degrees Odd

1992 ◽  
Vol 1 (4) ◽  
pp. 335-349 ◽  
Author(s):  
A. D. Scott
Keyword(s):  
Chromatic Number ◽  
Random Graphs ◽  
Connected Graph ◽  
Analogous Problem ◽  
Induced Subgraphs ◽  
Induced Subgraph

We prove that every connected graph of order n ≥ 2 has an induced subgraph with all degrees odd of order at least cn/log n, where cis a constant. We also give a bound in terms of chromatic number, and resolve the analogous problem for random graphs.

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