Salman Ghazal - Academia.edu (original) (raw)

Papers by Salman Ghazal

arXiv (Cornell University), Jun 10, 2015

We prove that the proof of existence of weighted local median order of weighted tournaments is wr... more We prove that the proof of existence of weighted local median order of weighted tournaments is wrong and that the proof of the correct statement which asserts that every digraph obtained from a tournament by deleting a set of arcs incident to the same vertex contains a mistake, in the paper entitled "Remarks on the second neighborhood problem". We introduce correct proofs of each.

Soit D un digraphe simple (sans cycle orienté de longueur 2 ). En 1990, P. Seymour a conjecturé q... more Soit D un digraphe simple (sans cycle orienté de longueur 2 ). En 1990, P. Seymour a conjecturé que D a un sommet v avec un second voisinage extérieur au moins aussi grand que son (premier) voisinage extérieur [1]. Cette conjecture est connue sous le nom de la conjecture du second voisinage du Seymour (SNC). Cette conjecture, si elle est vraie, impliquerait, un cas spécial plus faible (mais important) de la conjecture de Caccetta et Häggkvist [2] proposé en 1978 : tout digraphe D avec un degré extérieur minimum au moins égale à jV (D)j=k a un cycle orienté de longueur au plus k. Le cas particulier est k = 3, et le cas faible exige les deux : le degré extérieur minimum et le degré intérieur minimum de D sont au moins égaux à jV (D)j=k. La conjecture de Seymour restreinte au tournoi est connue sous le nom de conjecture de Dean [1]. En 1996, Fisher [3] a prouvé la conjecture de Dean en utilisant un argument de probabilité. En 2003, Chen, Shen et Yuster [4] ont démontré que tout digraph...

arXiv (Cornell University), Feb 27, 2016

Seymour's Second Neighborhood Conjecture asserts that every oriented graph has a vertex whose fir... more Seymour's Second Neighborhood Conjecture asserts that every oriented graph has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. Combs are the graphs having no induced C4, C4, C5, chair or chair. We characterize combs using dependency digraphs. We characterize the graphs having no induced C4, C4, chair or chair using dependency digraphs. Then we prove that every oriented graph missing a comb satisfies this conjecture. We then deduce that every oriented comb and every oriented threshold graph satisfies Seymour's conjecture.

arXiv (Cornell University), Oct 22, 2020

The celebrated Erdös-Hajnal conjecture states that for every undirected graph H there exists (H) ... more The celebrated Erdös-Hajnal conjecture states that for every undirected graph H there exists (H) > 0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or a stable set of size at least n (H). This conjecture has a directed equivalent version stating that for every tournament H there exists (H) > 0 such that every H-free n-vertex tournament T contains a transitive subtournament of order at least n (H). This conjecture is proved for few infinite families of tournaments. In this paper we construct a new infinite family of tournaments − the family of so-called flotilla-galaxies and we prove the correctness of the conjecture for every flotilla-galaxy tournament.

arXiv (Cornell University), Oct 21, 2020

A (2 + 1)-bispindle B(k1, k2; k3) is the union of two xy-dipaths of respective lengths k1 and k2,... more A (2 + 1)-bispindle B(k1, k2; k3) is the union of two xy-dipaths of respective lengths k1 and k2, and one yx-dipath of length k3, all these dipaths being pairwise internally disjoint. Recently, Cohen et al. conjectured that, for every positive integers k1, k2, k3, there is an integer g(k1, k2, k3) such that every strongly connected digraph not containing subdivisions of B(k1, k2; k3) has a chromatic number at most g(k1, k2, k3), and they proved it only for the case where k2 = 1. For Hamiltonian digraphs, we prove Cohen et al.'s conjecture, namely g(k1, k2, k3) ≤ 4k, where k = max{k1, k2, k3}. A two-blocks cycle C(k1, k2) is the union of two internally disjoint xy-dipaths of length k1 and k2 respectively. Addario et al. asked if the chromatic number of strong digraphs not containing subdivisions of a twoblocks cycle C(k1, k2) can be bounded from above by O(k1 + k2), which remains an open problem. Assuming that k = max{k1, k2}, the best reached upper bound, found by Kim et al., is 12k 2. In this article, we conjecture that this bound can be slightly improved to 4k 2 and we confirm our conjecture for some particular cases. Moreover, we provide a positive answer to Addario et al.'s question for the class of digraphs having a Hamiltonian directed path.

arXiv (Cornell University), Oct 22, 2020

A celebrated unresolved conjecture of Erdös and Hajnal states that for every undirected graph H t... more A celebrated unresolved conjecture of Erdös and Hajnal states that for every undirected graph H there exists (H) > 0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or a stable set of size at least n (H). The conjecture has a directed equivalent version stating that for every tournament H there exists (H) > 0 such that every H−free n−vertex tournament T contains a transitive subtournament of order at least n (H). Both the directed and the undirected versions of the conjecture are known to be true for small graphs (tournaments). So far the conjecture was proved only for some specific families of prime tournaments, tournaments constructed according to the so−called substitution procedure allowing to build bigger graphs, and for all five−vertex tournaments. Recently the conjecture was proved for all six−vertex tournament, with one exception, but the question about the correctness of the conjecture for all seven−vertex tournaments remained open. In this paper we prove the correctness of the conjecture for several seven−vertex tournaments.

Journal of Graph Theory

Erdös–Hajnal conjecture states that for every undirected graph there exists such that every undir... more Erdös–Hajnal conjecture states that for every undirected graph there exists such that every undirected graph on vertices that does not contain as an induced subgraph contains a clique or a stable set of size at least . This conjecture has a directed equivalent version stating that for every tournament there exists such that every ‐free ‐vertex tournament contains a transitive subtournament of order at least . This conjecture is known to hold for a few infinite families of tournaments. In this article we construct two new infinite families of tournaments—the family of so‐called galaxies with spiders and the family of so‐called asterisms, and we prove the correctness of the conjecture for these two families.

Seymour's Second Neighborhood Conjecture asserts that every digraph (without digons) has a vertex... more Seymour's Second Neighborhood Conjecture asserts that every digraph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. We prove its weighted version for tournaments missing a generalized star. As a consequence the weighted version holds for tournaments missing a sun, star, or a complete graph.

Seymour's Second Neighborhood Conjecture asserts that every digraph (without digons) has a vertex... more Seymour's Second Neighborhood Conjecture asserts that every digraph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. It is proved for tournaments, tournaments missing a matching and tournaments missing a generalized star. We prove this conjecture for classes of digraphs whose missing graph is a comb, a complete graph minus 2 independent edges, or a complete graph minus the edges of a cycle of length 5.

Soit D un digraphe simple (sans cycle orienté de longueur 2). En 1990, P. Seymour a conjecturé qu... more Soit D un digraphe simple (sans cycle orienté de longueur 2). En 1990, P. Seymour a conjecturé que D a un sommet v avec un second voisinage extérieur au moins aussi grand que son (premier) voisinage extérieur [14]. Cette conjecture est connue sous le nom de la conjecture du second voisinage du Seymour (SNC). Cette conjecture, si elle est vraie, impliquerait, un cas spécial plus faible (mais important) de la conjecture de Caccetta et Häggkvist [11] proposé en 1978: tout digraphe D avec un degré extérieur minimum au moins égale à |V (D)|/k a une cycle orienté de longueur au plus k. Le cas particulier est k = 3, et le cas faible exige les deux: le degré extérieur minimum et le degré intérieur minimum de D sont au moins égaux à |V (D)|/k. La conjecture de Seymour restreinte au tournoi est connue sous le nom de conjecture de Dean [14]. En 1996, Fisher [4] a prouvé la conjecture de Dean en utilisant un argument de probabilité. En 2003, Chen, Shen et Yuster [8] ont démontré que tout digrap...

Electronic Journal of Graph Theory and Applications, 2018

We find the structure of graphs that have no C 4 , C 4 , C 5 , S 3 , chair and co-chair as induce... more We find the structure of graphs that have no C 4 , C 4 , C 5 , S 3 , chair and co-chair as induced subgraphs. Then we deduce the structure of the graphs having no induced C 4 , C 4 , S 3 , chair and co-chair and the structure of the graphs G having no induced C 4 , C 4 and such that every induced P 4 of G is contained in an induced C 5 of G.

arXiv: Combinatorics, 2020

Erd\"{o}s-Hajnal conjecture states that for every undirected graph HHH there exists $ \epsil... more Erd\"{o}s-Hajnal conjecture states that for every undirected graph HHH there exists $ \epsilon(H) > 0 $ such that every undirected graph on $ n $ vertices that does not contain HHH as an induced subgraph contains a clique or a stable set of size at least $ n^{\epsilon(H)} .Thisconjecturehasadirectedequivalentversionstatingthatforeverytournament. This conjecture has a directed equivalent version stating that for every tournament .ThisconjecturehasadirectedequivalentversionstatingthatforeverytournamentH$ there exists $ \epsilon(H) > 0 $ such that every H−H-H−free n−n-n−vertex tournament TTT contains a transitive subtournament of order at least $ n^{\epsilon(H)} .Thisconjectureisprovedwhen. This conjecture is proved when .ThisconjectureisprovedwhenH$ is a galaxy or a constellation and for all five$-$vertex tournaments and for all six$-$vertex tournaments except one. In this paper we prove the correctness of the conjecture for any flotilla-galaxy tournament. This generalizes results some previous results.

We introduce three new elementary short proofs of the famous König's theorem which characterizes ... more We introduce three new elementary short proofs of the famous König's theorem which characterizes bipartite graphs by absence of odd cycles.

Electronic Journal of Graph Theory and Applications, 2016

Let D be a digraph without digons. Seymours second neighborhood conjecture states that D has a ve... more Let D be a digraph without digons. Seymours second neighborhood conjecture states that D has a vertex v such that d + (v) ≤ d ++ (v). Under some conditions, we prove this conjecture for digraphs missing n disjoint stars. Weaker conditions are required when n = 2 or 3. In some cases we exhibit 2 such vertices.

Indonesian Journal of Combinatorics, 2017

We introduce four new elementary short proofs of the famous…

We introduce four new elementary short proofs of the famous König's theorem which characterizes b... more We introduce four new elementary short proofs of the famous König's theorem which characterizes bipartite graphs by absence of odd cycles.

Http Www Theses Fr, Dec 15, 2011

Dans la suite, on suppose que les digraphes ne contiennent pas de boucle ni digon. Chemin. Un che... more Dans la suite, on suppose que les digraphes ne contiennent pas de boucle ni digon. Chemin. Un chemin P est un graphe avec un ensemble des sommets {v 1 , ..., v n } et un ensemble d'arêtes v i v i+1 pour i<n. Un chemin noté par v 1 v 2 ...v n est dit un v 1 v n-chemin ou un chemin de v 1à v n. Chemin orienté. Un chemin orienté P est un digraphe avec V (P)= {v 1 , ..., v n } et E(D)={(v i ,v i+1),i < n}. Un tel chemin est dit v 1 v n-chemin orienté. Onécrit P = v 1 ...v n. Cycle. Un cycle C est un graphe de sommets {v 1 , ..., v n } et d'arêtes v i v i+1 pour i<nplus l'arête v n v 1 .O nécrit C = v 1 ...v n. Circuit. Un circuit C est un digraphe avec un ensemble de sommets 9 {v 1 , ..., v n } et d'arcs (v i ,v i+1) pour i<nplus l'arc (v n ,v 1). Onécrit C = v 1 ...v n. Connexe. Un graphe G est connexe si tous deux sommets sont liés par un chemin. Fortement connexe. Un digraphe D est fortement connexe si pour tous deux sommets x et y, il existe un xy-chemin orienté. Arbre. Un arbre est un graphe connexe sans cycle. Arboresence. Une arbresence sortante (resp. rentrante) est un arbre orienté tel que, tous les sommets, sauf exactement un sommet sont de degré intérieur 1 (resp. extérieur 1). Etoile. Uneétoile est un arbre formé par un sommets et ses voisins. Couplage. Un couplage est un ensemble d'arcs (ou d'arêtes) deuxà deux disjoints. Stable. Un stable est un ensemble de sommets deuxà deux non adjacents. Graphe complet. Un graphe est dit complet si xy ∈ E(G) pour tous 2 sommets distincts x et y de V (G). Tournoi. Un tournoi est une orientation d'un graphe complet. Triangle. Un triangle est une cycle ayant 3 sommets. Un triangle dans un digraphes est dit cyclique s'il est un circuit. Sinon, il est acyclique. Carré. Un carré est une cycle ayant 4 sommets exactement. Roi. Un roi dans un digraphe D est un sommet x tel que {x}∪N + D (x)∪ N ++ D (x)=V (D). Degrés de digraphe. Le degré minimum de D est δ D = min{d(x); x ∈ V (D)}. Le degré extérieur minimum de D est δ + D = min{d + (x); x ∈ V (D)}. Le degré extérieur maximum de D est max{d + (x); x ∈ V (D)}. Le degré intérieur minimum de D est δ − D = min{d − (x); x ∈ V (D)}.

arXiv (Cornell University), Jun 10, 2015

We prove that the proof of existence of weighted local median order of weighted tournaments is wr... more We prove that the proof of existence of weighted local median order of weighted tournaments is wrong and that the proof of the correct statement which asserts that every digraph obtained from a tournament by deleting a set of arcs incident to the same vertex contains a mistake, in the paper entitled "Remarks on the second neighborhood problem". We introduce correct proofs of each.

Soit D un digraphe simple (sans cycle orienté de longueur 2 ). En 1990, P. Seymour a conjecturé q... more Soit D un digraphe simple (sans cycle orienté de longueur 2 ). En 1990, P. Seymour a conjecturé que D a un sommet v avec un second voisinage extérieur au moins aussi grand que son (premier) voisinage extérieur [1]. Cette conjecture est connue sous le nom de la conjecture du second voisinage du Seymour (SNC). Cette conjecture, si elle est vraie, impliquerait, un cas spécial plus faible (mais important) de la conjecture de Caccetta et Häggkvist [2] proposé en 1978 : tout digraphe D avec un degré extérieur minimum au moins égale à jV (D)j=k a un cycle orienté de longueur au plus k. Le cas particulier est k = 3, et le cas faible exige les deux : le degré extérieur minimum et le degré intérieur minimum de D sont au moins égaux à jV (D)j=k. La conjecture de Seymour restreinte au tournoi est connue sous le nom de conjecture de Dean [1]. En 1996, Fisher [3] a prouvé la conjecture de Dean en utilisant un argument de probabilité. En 2003, Chen, Shen et Yuster [4] ont démontré que tout digraph...

arXiv (Cornell University), Feb 27, 2016

Seymour's Second Neighborhood Conjecture asserts that every oriented graph has a vertex whose fir... more Seymour's Second Neighborhood Conjecture asserts that every oriented graph has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. Combs are the graphs having no induced C4, C4, C5, chair or chair. We characterize combs using dependency digraphs. We characterize the graphs having no induced C4, C4, chair or chair using dependency digraphs. Then we prove that every oriented graph missing a comb satisfies this conjecture. We then deduce that every oriented comb and every oriented threshold graph satisfies Seymour's conjecture.

arXiv (Cornell University), Oct 22, 2020

The celebrated Erdös-Hajnal conjecture states that for every undirected graph H there exists (H) ... more The celebrated Erdös-Hajnal conjecture states that for every undirected graph H there exists (H) > 0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or a stable set of size at least n (H). This conjecture has a directed equivalent version stating that for every tournament H there exists (H) > 0 such that every H-free n-vertex tournament T contains a transitive subtournament of order at least n (H). This conjecture is proved for few infinite families of tournaments. In this paper we construct a new infinite family of tournaments − the family of so-called flotilla-galaxies and we prove the correctness of the conjecture for every flotilla-galaxy tournament.

arXiv (Cornell University), Oct 21, 2020

A (2 + 1)-bispindle B(k1, k2; k3) is the union of two xy-dipaths of respective lengths k1 and k2,... more A (2 + 1)-bispindle B(k1, k2; k3) is the union of two xy-dipaths of respective lengths k1 and k2, and one yx-dipath of length k3, all these dipaths being pairwise internally disjoint. Recently, Cohen et al. conjectured that, for every positive integers k1, k2, k3, there is an integer g(k1, k2, k3) such that every strongly connected digraph not containing subdivisions of B(k1, k2; k3) has a chromatic number at most g(k1, k2, k3), and they proved it only for the case where k2 = 1. For Hamiltonian digraphs, we prove Cohen et al.'s conjecture, namely g(k1, k2, k3) ≤ 4k, where k = max{k1, k2, k3}. A two-blocks cycle C(k1, k2) is the union of two internally disjoint xy-dipaths of length k1 and k2 respectively. Addario et al. asked if the chromatic number of strong digraphs not containing subdivisions of a twoblocks cycle C(k1, k2) can be bounded from above by O(k1 + k2), which remains an open problem. Assuming that k = max{k1, k2}, the best reached upper bound, found by Kim et al., is 12k 2. In this article, we conjecture that this bound can be slightly improved to 4k 2 and we confirm our conjecture for some particular cases. Moreover, we provide a positive answer to Addario et al.'s question for the class of digraphs having a Hamiltonian directed path.

arXiv (Cornell University), Oct 22, 2020

A celebrated unresolved conjecture of Erdös and Hajnal states that for every undirected graph H t... more A celebrated unresolved conjecture of Erdös and Hajnal states that for every undirected graph H there exists (H) > 0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or a stable set of size at least n (H). The conjecture has a directed equivalent version stating that for every tournament H there exists (H) > 0 such that every H−free n−vertex tournament T contains a transitive subtournament of order at least n (H). Both the directed and the undirected versions of the conjecture are known to be true for small graphs (tournaments). So far the conjecture was proved only for some specific families of prime tournaments, tournaments constructed according to the so−called substitution procedure allowing to build bigger graphs, and for all five−vertex tournaments. Recently the conjecture was proved for all six−vertex tournament, with one exception, but the question about the correctness of the conjecture for all seven−vertex tournaments remained open. In this paper we prove the correctness of the conjecture for several seven−vertex tournaments.

Journal of Graph Theory

Erdös–Hajnal conjecture states that for every undirected graph there exists such that every undir... more Erdös–Hajnal conjecture states that for every undirected graph there exists such that every undirected graph on vertices that does not contain as an induced subgraph contains a clique or a stable set of size at least . This conjecture has a directed equivalent version stating that for every tournament there exists such that every ‐free ‐vertex tournament contains a transitive subtournament of order at least . This conjecture is known to hold for a few infinite families of tournaments. In this article we construct two new infinite families of tournaments—the family of so‐called galaxies with spiders and the family of so‐called asterisms, and we prove the correctness of the conjecture for these two families.

Seymour's Second Neighborhood Conjecture asserts that every digraph (without digons) has a vertex... more Seymour's Second Neighborhood Conjecture asserts that every digraph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. We prove its weighted version for tournaments missing a generalized star. As a consequence the weighted version holds for tournaments missing a sun, star, or a complete graph.

Seymour's Second Neighborhood Conjecture asserts that every digraph (without digons) has a vertex... more Seymour's Second Neighborhood Conjecture asserts that every digraph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. It is proved for tournaments, tournaments missing a matching and tournaments missing a generalized star. We prove this conjecture for classes of digraphs whose missing graph is a comb, a complete graph minus 2 independent edges, or a complete graph minus the edges of a cycle of length 5.

Soit D un digraphe simple (sans cycle orienté de longueur 2). En 1990, P. Seymour a conjecturé qu... more Soit D un digraphe simple (sans cycle orienté de longueur 2). En 1990, P. Seymour a conjecturé que D a un sommet v avec un second voisinage extérieur au moins aussi grand que son (premier) voisinage extérieur [14]. Cette conjecture est connue sous le nom de la conjecture du second voisinage du Seymour (SNC). Cette conjecture, si elle est vraie, impliquerait, un cas spécial plus faible (mais important) de la conjecture de Caccetta et Häggkvist [11] proposé en 1978: tout digraphe D avec un degré extérieur minimum au moins égale à |V (D)|/k a une cycle orienté de longueur au plus k. Le cas particulier est k = 3, et le cas faible exige les deux: le degré extérieur minimum et le degré intérieur minimum de D sont au moins égaux à |V (D)|/k. La conjecture de Seymour restreinte au tournoi est connue sous le nom de conjecture de Dean [14]. En 1996, Fisher [4] a prouvé la conjecture de Dean en utilisant un argument de probabilité. En 2003, Chen, Shen et Yuster [8] ont démontré que tout digrap...

Electronic Journal of Graph Theory and Applications, 2018

We find the structure of graphs that have no C 4 , C 4 , C 5 , S 3 , chair and co-chair as induce... more We find the structure of graphs that have no C 4 , C 4 , C 5 , S 3 , chair and co-chair as induced subgraphs. Then we deduce the structure of the graphs having no induced C 4 , C 4 , S 3 , chair and co-chair and the structure of the graphs G having no induced C 4 , C 4 and such that every induced P 4 of G is contained in an induced C 5 of G.

arXiv: Combinatorics, 2020

Erd\"{o}s-Hajnal conjecture states that for every undirected graph HHH there exists $ \epsil... more Erd\"{o}s-Hajnal conjecture states that for every undirected graph HHH there exists $ \epsilon(H) > 0 $ such that every undirected graph on $ n $ vertices that does not contain HHH as an induced subgraph contains a clique or a stable set of size at least $ n^{\epsilon(H)} .Thisconjecturehasadirectedequivalentversionstatingthatforeverytournament. This conjecture has a directed equivalent version stating that for every tournament .ThisconjecturehasadirectedequivalentversionstatingthatforeverytournamentH$ there exists $ \epsilon(H) > 0 $ such that every H−H-H−free n−n-n−vertex tournament TTT contains a transitive subtournament of order at least $ n^{\epsilon(H)} .Thisconjectureisprovedwhen. This conjecture is proved when .ThisconjectureisprovedwhenH$ is a galaxy or a constellation and for all five$-$vertex tournaments and for all six$-$vertex tournaments except one. In this paper we prove the correctness of the conjecture for any flotilla-galaxy tournament. This generalizes results some previous results.

We introduce three new elementary short proofs of the famous König's theorem which characterizes ... more We introduce three new elementary short proofs of the famous König's theorem which characterizes bipartite graphs by absence of odd cycles.

Electronic Journal of Graph Theory and Applications, 2016

Let D be a digraph without digons. Seymours second neighborhood conjecture states that D has a ve... more Let D be a digraph without digons. Seymours second neighborhood conjecture states that D has a vertex v such that d + (v) ≤ d ++ (v). Under some conditions, we prove this conjecture for digraphs missing n disjoint stars. Weaker conditions are required when n = 2 or 3. In some cases we exhibit 2 such vertices.

Indonesian Journal of Combinatorics, 2017

We introduce four new elementary short proofs of the famous…

We introduce four new elementary short proofs of the famous König's theorem which characterizes b... more We introduce four new elementary short proofs of the famous König's theorem which characterizes bipartite graphs by absence of odd cycles.

Http Www Theses Fr, Dec 15, 2011

Dans la suite, on suppose que les digraphes ne contiennent pas de boucle ni digon. Chemin. Un che... more Dans la suite, on suppose que les digraphes ne contiennent pas de boucle ni digon. Chemin. Un chemin P est un graphe avec un ensemble des sommets {v 1 , ..., v n } et un ensemble d'arêtes v i v i+1 pour i<n. Un chemin noté par v 1 v 2 ...v n est dit un v 1 v n-chemin ou un chemin de v 1à v n. Chemin orienté. Un chemin orienté P est un digraphe avec V (P)= {v 1 , ..., v n } et E(D)={(v i ,v i+1),i < n}. Un tel chemin est dit v 1 v n-chemin orienté. Onécrit P = v 1 ...v n. Cycle. Un cycle C est un graphe de sommets {v 1 , ..., v n } et d'arêtes v i v i+1 pour i<nplus l'arête v n v 1 .O nécrit C = v 1 ...v n. Circuit. Un circuit C est un digraphe avec un ensemble de sommets 9 {v 1 , ..., v n } et d'arcs (v i ,v i+1) pour i<nplus l'arc (v n ,v 1). Onécrit C = v 1 ...v n. Connexe. Un graphe G est connexe si tous deux sommets sont liés par un chemin. Fortement connexe. Un digraphe D est fortement connexe si pour tous deux sommets x et y, il existe un xy-chemin orienté. Arbre. Un arbre est un graphe connexe sans cycle. Arboresence. Une arbresence sortante (resp. rentrante) est un arbre orienté tel que, tous les sommets, sauf exactement un sommet sont de degré intérieur 1 (resp. extérieur 1). Etoile. Uneétoile est un arbre formé par un sommets et ses voisins. Couplage. Un couplage est un ensemble d'arcs (ou d'arêtes) deuxà deux disjoints. Stable. Un stable est un ensemble de sommets deuxà deux non adjacents. Graphe complet. Un graphe est dit complet si xy ∈ E(G) pour tous 2 sommets distincts x et y de V (G). Tournoi. Un tournoi est une orientation d'un graphe complet. Triangle. Un triangle est une cycle ayant 3 sommets. Un triangle dans un digraphes est dit cyclique s'il est un circuit. Sinon, il est acyclique. Carré. Un carré est une cycle ayant 4 sommets exactement. Roi. Un roi dans un digraphe D est un sommet x tel que {x}∪N + D (x)∪ N ++ D (x)=V (D). Degrés de digraphe. Le degré minimum de D est δ D = min{d(x); x ∈ V (D)}. Le degré extérieur minimum de D est δ + D = min{d + (x); x ∈ V (D)}. Le degré extérieur maximum de D est max{d + (x); x ∈ V (D)}. Le degré intérieur minimum de D est δ − D = min{d − (x); x ∈ V (D)}.