### Book chapter

## Логико-термальная эквивалентность программ с динамической памятью

### In book

The study aimed to test measurement invariance of the Russian-language EmIn questionnaire (by D. Lyusin) for emotional intelligence assessment in two samples, from Russia (n = 275) and Azerbaijan (n = 275). Exploratory factor analysis on pooled sample revealed a 4-factor structure with dimensions interpreted as understanding of one’s own emotions, management of one’s own emotions, understanding of others’ emotions, management of others’ emotions. Using confirmatory factor analysis, strong factorial invariance (equivalence of factor loadings and intercepts) was established, which allows to compare means scores in two cultures. Russians, compared to the Azerbaijani, report better understanding of one’s own emotions and management of one’s own emotions. Russian males report better management of their own emotions, compared to Russian females (in all age groups). Azerbaijani females report better understanding of others’ emotions, compared to Azerbaijani males (except for the senior age group). The results are interpreted based on existing knowledge of cross-cultural differences between Russian and Azerbaijan in cultural values, such as individualism and masculinity.

We say that a graph *G* has the CIS-property and call it a CIS-graph if every maximal clique and every maximal stable set of *G* intersects.

By definition, *G* is a CIS-graph if and only if the complementary graph \(\overline {G}\) is a CIS-graph. Let us substitute a vertex *v* of a graph *G′* by a graph *G*″ and denote the obtained graph by *G*. It is also easy to see that *G* is a CIS-graph if and only if both *G′* and *G*″ are CIS-graphs. In other words, CIS-graphs respect complementation and substitution. Yet, this class is not hereditary, that is, an induced subgraph of a CIS-graph may have no CIS-property. Perhaps, for this reason, the problems of efficient characterization and recognition of CIS-graphs are difficult and remain open. In this paper we only give some necessary and some sufficient conditions for the CIS-property to hold.

There are obvious sufficient conditions. It is known that *P*4-free graphs have the CIS-property and it is easy to see that *G* is a CIS-graph whenever each maximal clique of *G* has a simplicial vertex. However, these conditions are not necessary.

There are also simple necessary conditions. Given an integer *k* ≥ 2, a *comb* (or *k*-*comb*) *S**k* is a graph with 2*k* vertices *k* of which, *v*1, …, *v**k*, form a clique *C*, while others, \(v^{\prime }_1, \ldots , v^{\prime }_k,\) form a stable set *S*, and \((v_i,v^{\prime }_i)\) is an edge for all *i* = 1, …, *k*, and there are no other edges. The complementary graph \(\overline {S_k}\) is called an *anti-comb* (or *k*-anti-comb). Clearly, *S* and *C* switch in the complementary graphs. Obviously, the combs and anti-combs are not CIS-graphs, since *C* ∩ *S* = ∅. Hence, if a CIS-graph *G* contains an induced comb or anti-comb, then it must be settled, that is, *G* must contain a vertex *v* connected to all vertices of *C* and to no vertex of *S*. For *k* = 2 this observation was made by Claude Berge in 1985. However, these conditions are only necessary.

The following sufficient conditions are more difficult to prove: *G* is a CIS-graph whenever *G*contains no induced 3-combs and 3-anti-combs, and every induced 2-comb is settled in *G*, as it was conjectured by Vasek Chvatal in early 90s. First partial results were published by his student Wenan Zang from Rutgers Center for Operations Research. Then, the statement was proven by Deng, Li, and Zang. Here we give an alternative proof, which is of independent interest; it is based on some properties of the product of two Petersen graphs.

It is an open question whether *G* is a CIS-graph if it contains no induced 4-combs and 4-anti-combs, and all induced 3-combs, 3-anti-combs, and 2-combs are settled in *G*.

We generalize the concept of CIS-graphs as follows. For an integer *d* ≥ 2 we define a *d*-*graph* \(\mathcal {G} = (V; E_1, \ldots , E_d)\) as a complete graph whose edges are colored by *d*colors (that is, partitioned into *d* sets). We say that \(\mathcal {G}\) is a CIS-*d*-graph (has the CIS-*d*-property) if \(\bigcap _{i=1}^d C_i \neq \emptyset \) whenever for each *i* = 1, …, *d* the set *C**i* is a maximal color *i*-free subset of *V* , that is, (*v*, *v′*)∉*E**i* for any *v*, *v′*∈ *C**i*. Clearly, in case *d* = 2 we return to the concept of CIS-graphs. (More accurately, CIS-2-graph is a pair of two complementary CIS-graphs.) We conjecture that each CIS-*d*-graph is a Gallai graph, that is, it contains no triangle colored by 3 distinct colors. We obtain results supporting this conjecture and also show that if it holds then characterization and recognition of CIS-*d*-graphs are easily reduced to characterization and recognition of CIS-graphs.

We also prove the following statement. Let \(\mathcal {G} = (V; E_1, \ldots , E_d)\) be a Gallai *d*-graph such that at least *d* − 1 of its *d* chromatic components are CIS-graphs, then \(\mathcal {G}\) has the CIS-*d*-property. In particular, the remaining chromatic component of \(\mathcal {G}\) is a CIS-graph too. Moreover, all 2*d* unions of *d* chromatic components of \(\mathcal {G}\) are CIS-graphs.

This proceedings set contains 85 selected full papers presentedat the 3rd International Conference on Modelling, Computation and Optimization in Information Systems and Management Sciences - MCO 2015, held on May 11–13, 2015 at Lorraine University, France. The present part II of the 2 volume set includes articles devoted to Data analysis and Data mining, Heuristic / Meta heuristic methods for operational research applications, Optimization applied to surveillance and threat detection, Maintenance and Scheduling, Post Crises banking and eco-finance modelling, Transportation, as well as Technologies and methods for multi-stakeholder decision analysis in public settings.