Imed Zaguia
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Imed Zaguia has a total of 2 co-authors. Here are the top co-authors by number of shared publications:
- Maurice Pouzet
Department University of Calgary
5 shared publications - Alexander Clow
Department of Mathematics Simon Fraser University Burnaby BC Canada
1 shared publication
Latest Publications
- Open Access
Cops and Robbers, Clique Covers, and Induced Cycles
We consider the Cops and Robbers game played on finite simple graphs. In a graph $G$, the number of cops required to capture a robber in the Cops and Robbers game is denoted by $c(G)$. For all graphs $G$, $c(G) \leq α(G) \leq θ(G)$ where $α(G)$ and $θ(G)$ are the independence number and clique cover number respectively. In 2022 Turcotte asked if $c(G) < α(G)$ for all graphs with $α(G) \geq 3$. Recently, Char, Maniya, and Pradhan proved this is false, at least when $α= 3$,by demonstrating the compliment of the Shrikhande graph has cop number and independence number $3$. We prove, using random graphs, the stronger result that for all $k\geq 1$ there exists a graph $G$ such that $c(G) = α(G) = θ(G) = k$. Next, we consider the structure of graphs with $c(G) = θ(G) \geq 3$. We prove, using structural arguments, that any graphs $G$ which satisfies $c(G) = θ(G) = k \geq 3$ contain induced cycles of all lengths $3\leq t \leq k+1$. This implies all perfect graphs $G$ with $α(G)\geq 4$ have $c(G) < α(G)$. Additionally,we discuss if typical triangle-free and $C_4$-free graphs will have $c(G) < α(G)$.Open AccessCops and Robbers, Clique Covers, and Induced Cycles
We consider the Cops and Robbers game played on finite simple graphs. In a graph $G$, the number of cops required to capture a robber in the Cops and Robbers game is denoted by $c(G)$. For all graphs $G$, $c(G) \leq α(G) \leq θ(G)$ where $α(G)$ and $θ(G)$ are the independence number and clique cover number respectively. In 2022 Turcotte asked if $c(G) < α(G)$ for all graphs with $α(G) \geq 3$. Recently, Char, Maniya, and Pradhan proved this is false, at least when $α= 3$,by demonstrating the compliment of the Shrikhande graph has cop number and independence number $3$. We prove, using random graphs, the stronger result that for all $k\geq 1$ there exists a graph $G$ such that $c(G) = α(G) = θ(G) = k$. Next, we consider the structure of graphs with $c(G) = θ(G) \geq 3$. We prove, using structural arguments, that any graphs $G$ which satisfies $c(G) = θ(G) = k \geq 3$ contain induced cycles of all lengths $3\leq t \leq k+1$. This implies all perfect graphs $G$ with $α(G)\geq 4$ have $c(G) < α(G)$. Additionally,we discuss if typical triangle-free and $C_4$-free graphs will have $c(G) < α(G)$. - Open Access
The structure of interval orders with no infinite antichain
We prove that every interval order $P$ with no infinite antichain has a Gallai decomposition. That is, $P$ is a lexicographical sum of proper interval orders over a chain, an antichain or a prime interval order. This is a consequence of the fact that the tree decomposition of a graph into robust modules, as introduced by Courcelle and Delhommé (Theoretical Computer Science \textbf{394} (2008) 1--38), is chain finite whenever the graph has no infinite independent sets. Next, we prove that every prime interval order with no infinite antichain is at most countable and scattered. Furthermore, for each countable ordinal $α$ we exhibit an example of a well-quasi-ordered prime interval order $P_α$ whose chain of maximal antichains has Hausdorff rank $α$.Open AccessThe structure of interval orders with no infinite antichain
We prove that every interval order $P$ with no infinite antichain has a Gallai decomposition. That is, $P$ is a lexicographical sum of proper interval orders over a chain, an antichain or a prime interval order. This is a consequence of the fact that the tree decomposition of a graph into robust modules, as introduced by Courcelle and Delhommé (Theoretical Computer Science \textbf{394} (2008) 1--38), is chain finite whenever the graph has no infinite independent sets. Next, we prove that every prime interval order with no infinite antichain is at most countable and scattered. Furthermore, for each countable ordinal $α$ we exhibit an example of a well-quasi-ordered prime interval order $P_α$ whose chain of maximal antichains has Hausdorff rank $α$. - 18 June 2024
- Published by Elsevier in Discrete Mathematics
- Vol. 347 (10)
Some progress on the Aharoni–Korman conjecture
- 18 June 2024
- Published by Elsevier in Discrete Mathematics
- Vol. 347 (10)
Some progress on the Aharoni–Korman conjecture
- 3 October 2023
- Published by Elsevier in European Journal of Combinatorics
- Vol. 119
Hereditary classes of ordered sets of width at most two
- 3 October 2023
- Published by Elsevier in European Journal of Combinatorics
- Vol. 119
Hereditary classes of ordered sets of width at most two
- Open Access
The Aharoni--Korman conjecture for posets whose incomparability graph is locally finite
Aharoni and Korman (Order 9 (1992) 245--253) have conjectured that every ordered set without infinite antichains possesses a chain and a partition into antichains so that each part intersects the chain. The conjecture is verified for posets whose incomparability graph is locally finite. It follows that the conjecture is true for $(3 + 1)$-free posets with no infinite antichains.
Open AccessThe Aharoni--Korman conjecture for posets whose incomparability graph is locally finite
Aharoni and Korman (Order 9 (1992) 245--253) have conjectured that every ordered set without infinite antichains possesses a chain and a partition into antichains so that each part intersects the chain. The conjecture is verified for posets whose incomparability graph is locally finite. It follows that the conjecture is true for $(3 + 1)$-free posets with no infinite antichains.
- Open Access
The Aharoni--Korman conjecture for $N$-free posets with no infinite antichain
We give a necessary and sufficient condition for a $P_4$-free graph to be a cograph. This allows us to obtain a simple proof of the fact that finite $P_4$-free graphs are finite cographs. We also prove that $N$-free chain complete posets and $N$-free posets with no infinite antichains are series-parallel. As a consequence, we obtain that every $N$-free poset with no infinite antichain has a chain and a partition into antichains so that each part intersects the chain. This answers a conjecture of Aharoni and Korman (Order \textbf{9} (1992) 245--253) in this case.
Open AccessThe Aharoni--Korman conjecture for $N$-free posets with no infinite antichain
We give a necessary and sufficient condition for a $P_4$-free graph to be a cograph. This allows us to obtain a simple proof of the fact that finite $P_4$-free graphs are finite cographs. We also prove that $N$-free chain complete posets and $N$-free posets with no infinite antichains are series-parallel. As a consequence, we obtain that every $N$-free poset with no infinite antichain has a chain and a partition into antichains so that each part intersects the chain. This answers a conjecture of Aharoni and Korman (Order \textbf{9} (1992) 245--253) in this case.
- Open Access
Hereditary classes of ordered sets of width at most two
This paper is a contribution to the study of hereditary classes of relational structures, these classes being quasi-ordered by embeddability. It deals with the specific case of ordered sets of width two and the corresponding bichains and incomparability graphs. Several open problems about hereditary classes of relational structures which have been considered over the years have positive answer in this case. For example, well-quasi-ordered hereditary classes of finite bipartite permutation graphs, respectively finite 321-avoiding permutations, have been characterized by Korpelainen, Lozin and Mayhill, respectively by Albert, Brignall, Ru\v{s}kuc and Vatter. We provide another proof of the results mentioned above. It is based on the existence of a countable universal poset of width two, obtained by the first author in 1978, his notion of multichainability (1978) (a kind of analog to letter-graphs), and metric properties of incomparability graphs. Using Laver's theorem (1971) on better-quasi-ordering (bqo) of countable chains we prove that a wqo hereditary class of finite or countable bipartite permutation graphs is necessarily bqo. This gives a positive answer to a conjecture of Nash-Williams (1965) in this case. We extend a previous result of Albert et al. by proving that if a hereditary class of finite, respectively countable, bipartite permutation graphs is wqo, respectively bqo, then the corresponding hereditary classes of posets of width at most two and bichains are wqo, respectively bqo. Several notions of labelled wqo are also considered. We prove that they are all equivalent in the case of bipartite permutation graphs, posets of width at most two and the corresponding bichains. We characterize hereditary classes of finite bipartite permutation graphs which remain wqo when labels from a wqo are added.
Open AccessHereditary classes of ordered sets of width at most two
This paper is a contribution to the study of hereditary classes of relational structures, these classes being quasi-ordered by embeddability. It deals with the specific case of ordered sets of width two and the corresponding bichains and incomparability graphs. Several open problems about hereditary classes of relational structures which have been considered over the years have positive answer in this case. For example, well-quasi-ordered hereditary classes of finite bipartite permutation graphs, respectively finite 321-avoiding permutations, have been characterized by Korpelainen, Lozin and Mayhill, respectively by Albert, Brignall, Ru\v{s}kuc and Vatter. We provide another proof of the results mentioned above. It is based on the existence of a countable universal poset of width two, obtained by the first author in 1978, his notion of multichainability (1978) (a kind of analog to letter-graphs), and metric properties of incomparability graphs. Using Laver's theorem (1971) on better-quasi-ordering (bqo) of countable chains we prove that a wqo hereditary class of finite or countable bipartite permutation graphs is necessarily bqo. This gives a positive answer to a conjecture of Nash-Williams (1965) in this case. We extend a previous result of Albert et al. by proving that if a hereditary class of finite, respectively countable, bipartite permutation graphs is wqo, respectively bqo, then the corresponding hereditary classes of posets of width at most two and bichains are wqo, respectively bqo. Several notions of labelled wqo are also considered. We prove that they are all equivalent in the case of bipartite permutation graphs, posets of width at most two and the corresponding bichains. We characterize hereditary classes of finite bipartite permutation graphs which remain wqo when labels from a wqo are added.
- Open Access
Metric properties of incomparability graphs with an emphasis on paths
We describe some metric properties of incomparability graphs. We consider the problem of the existence of infinite paths, either induced or isometric, in the incomparability graph of a poset. Among other things, we show that if the incomparability graph of a poset is connected and has infinite diameter, then it contains an infinite induced path. Furthermore, if the diameter of the set of vertices of degree at least $3$ is infinite, then the graph contains as an induced subgraph either a comb or a kite.Open AccessMetric properties of incomparability graphs with an emphasis on paths
We describe some metric properties of incomparability graphs. We consider the problem of the existence of infinite paths, either induced or isometric, in the incomparability graph of a poset. Among other things, we show that if the incomparability graph of a poset is connected and has infinite diameter, then it contains an infinite induced path. Furthermore, if the diameter of the set of vertices of degree at least $3$ is infinite, then the graph contains as an induced subgraph either a comb or a kite. - Open Access
Interval orders, semiorders and ordered groups
We prove that the order of an ordered group is an interval order if and only if it is a semiorder. Next, we prove that every semiorder is isomorphic to a collection $\mathcal J$ of intervals of some totally ordered abelian group, these intervals being of the form $[x, x+ \alpha[$ for some positive $\alpha$. We describe ordered groups such that the ordering is a semiorder and we introduce threshold groups generalizing totally ordered groups. We show that the free group on finitely many generators and the Thompson group $\mathbb F$ can be equipped with a compatible semiorder which is not a weak order. On another hand, a group introduced by Clifford cannot.Open AccessInterval orders, semiorders and ordered groups
We prove that the order of an ordered group is an interval order if and only if it is a semiorder. Next, we prove that every semiorder is isomorphic to a collection $\mathcal J$ of intervals of some totally ordered abelian group, these intervals being of the form $[x, x+ \alpha[$ for some positive $\alpha$. We describe ordered groups such that the ordering is a semiorder and we introduce threshold groups generalizing totally ordered groups. We show that the free group on finitely many generators and the Thompson group $\mathbb F$ can be equipped with a compatible semiorder which is not a weak order. On another hand, a group introduced by Clifford cannot.