The Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods. The rst of these three approaches yielded the rst (and to date only) proof of the SPGC; the other two remain promising to consider...

A graph is perfect if, in all its induced subgraphs, the size of a largest clique is equal to the chromatic number. Examples of perfect graphs include bipartite graphs, line graphs of bipartite graphs and the complements of such graphs. These four classes of perfect graphs will be called basic. In 1960, Berge formulated two conjectures about perfect graphs, one stronger than the other. The weak...

the permit to work system (ptws) provides a practical framework for our staff and contractors to abide by our commitment of no harm to people or the environment and also, the ptw system intended to provide an effective means of identifying significant hazards and managing significant hse risk activities. in other words, it provides a set of eyes that focus on significant hse risk activities to ...

We prove that a minimal imperfect graph containing a vertex which is not on any P 5 has no odd pair. A graph is perfect if the vertices of any induced subgraph H can be colored, in such a way that no two adjacent vertices receive the same color, with a number of colors (denoted by (H)) not exceeding the cardinality !(H) of a maximum clique of H. A graph is minimal imperfect if all its proper in...

For terms not defined here, the reader is referred to [6]. For a graph G=(V, E), we denote by G the complement of G. We use |(G) to denote the size of a largest clique in G and :(G) to denote the size of a largest stable set in G, or simply | and : when no confusion is possible. A graph G is said to be perfect if, for each induced subgraph H of G, H can be coloured with |(H) colours such that e...

We prove that a 2K 2 minimal imperfect Berge graph (if any) contains a K 6 : A graph is perfect if the vertices of any induced subgraph H can be colored, in such a way that no two adjacent vertices receive the same color, with a number of colors (denoted by (H)) not exceeding the cardinality !(H) of a maximum clique of H. A graph is minimal imperfect if all its proper induced subgraphs are perf...

The present research introduces adsorption of sulfonated cobalt phthalocyanine catalyst (Europhtal) from the real wastewater that is produced in mercaptan removal unit (Merox) third refinery, South Pars Gas Complex (SPGC), Assaluyeh, Iran. Therefore, novel low-cost natural zeolite modified with polyaluminium chloride (PAC) was investigated as an adsorbent for treatment Europhtal wastewater. Cen...

Combined heat and power systems are becoming more and more important, regarding their enhanced efficiency, energy saving, and environmental aspects. In the peresent study, three configurations of combined heat and power systems are intended as an alternative to separate production plant by considering environmental aspects. First and second laws of thermodynamics are adapted to the operatin...

A free Plain Language Summary can be found within the Supporting Information of this article. Our planet is losing biodiversity at unprecedented rates due to land-use change, direct exploitation, climate pollution and invasion exotic species (Cardinale et al., 2012; IPBES, 2019; Millennium Ecosystem Assessment, 2005; Tilman, 1999). In Western countries, for example, loss had started already in ...